**Explanation:** The SI unit of work is the joule, defined as the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.

**Explanation:** Kinetic energy (KE) is given by the formula KE = (1/2)mv², indicating it is directly proportional to the mass (m) and the square of the velocity (v) of an object.

**Explanation:** Work is defined as the product of the force and the displacement in the direction of the force. If there is no displacement, the work done is zero, regardless of the force applied.

**Explanation:** Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. It is given by the formula U = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above the reference point.

**Explanation:** Power is the rate at which work is done or energy is transferred. It is measured in watts (W), where one watt equals one joule per second.

**Explanation:** Non-conservative forces, such as friction, cause energy dissipation (usually in the form of heat), and the work done by these forces depends on the path taken, not just the initial and final positions.

**Explanation:** The total mechanical energy of a system is the sum of its kinetic energy (KE) and potential energy (PE). It is a conserved quantity in the absence of non-conservative forces.

**Explanation:** The law of conservation of energy states that the total energy of an isolated system remains constant over time, implying that energy cannot be created or destroyed, only transformed from one form to another.

**Explanation:** A body is in equilibrium if the sum of all the forces acting on it is zero, resulting in no acceleration. This can be static equilibrium (at rest) or dynamic equilibrium (moving with constant velocity).

**Explanation:** Power is the rate of doing work. If a force is applied to move an object with a certain velocity, power can be calculated using the formula P = F × v, where F is the force and v is the velocity.

**Explanation:** In physics, work is defined as the product of the force applied to an object and the displacement of the object in the direction of the force. Mathematically, it is expressed as \( W = F \times d \times \cos(\theta) \), where \( \theta \) is the angle between the force and the displacement vector.

**Explanation:** For work to be done on an object, there must be a displacement of the object. If the object does not move, no work is done, regardless of the amount of force applied.

**Explanation:** When the angle between the force and displacement is 90 degrees, the work done is zero because \( \cos(90^\circ) = 0 \). This means the force does not contribute to the displacement in the direction of the force.

**Explanation:** When a force is applied at an angle \( \theta \) to the direction of displacement, the work done is calculated using \( W = F \times d \times \cos(\theta) \), where \( \theta \) is the angle between the force and the displacement vectors.

**Explanation:** The work done is calculated using the formula \( W = F \times d \). Here, \( F = 10 \) N and \( d = 5 \) m, so \( W = 10 \times 5 = 50 \) Joules.

**Explanation:** No work is done on the wall because there is no displacement. Work is only done when a force causes displacement in the direction of the force.

**Explanation:** When an object falls freely under the influence of gravity, the work done by gravity is positive because the force of gravity and the displacement of the object are in the same direction.

**Explanation:** Only the component of the force that is in the direction of the displacement contributes to the work done. This is calculated as \( F \cos(\theta) \), where \( \theta \) is the angle between the force and the displacement.

**Explanation:** Negative work occurs when the force applied to an object is in the opposite direction to its displacement. This means the force is resisting the movement, such as friction or air resistance.

**Explanation:** The work done by the force is negative because the force and displacement are in opposite directions. Using the formula \( W = F \times d \times \cos(180^\circ) \), where \( \cos(180^\circ) = -1 \), we get \( W = 5 \times 4 \times -1 = -20 \) Joules.

**Explanation:** Kinetic energy is the energy possessed by an object due to its motion. It is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.

**Explanation:** Potential energy is the energy stored in an object due to its position or state. Water held behind a dam has gravitational potential energy due to its elevated position.

**Explanation:** Thermal energy, also known as heat energy, is the energy that comes from the temperature of matter. The faster the particles in an object move, the more thermal energy they produce.

**Explanation:** Chemical energy is the energy stored in the bonds of chemical compounds, such as molecules and atoms. This energy is released during chemical reactions.

**Explanation:** A compressed spring has elastic potential energy due to its deformation. This energy can be released when the spring returns to its original shape.

**Explanation:** Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is given by the formula \( U = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground.

**Explanation:** Thermal energy is the total kinetic energy of the particles in a substance due to their random motion. It is related to the temperature of the substance.

**Explanation:** In a battery-powered flashlight, chemical energy stored in the battery is converted to electrical energy, which then powers the light bulb to produce light energy.

**Explanation:** At the highest point of its swing, a pendulum has maximum potential energy and minimum kinetic energy. As it swings down, potential energy is converted into kinetic energy.

**Explanation:** The law of conservation of energy states that the total energy of an isolated system remains constant over time. Energy can be transformed from one form to another but cannot be created or destroyed.

**Explanation:** Kinetic energy is the energy an object possesses due to its motion. A rolling ball has kinetic energy because it is moving.

**Explanation:** When wood burns in a campfire, the chemical energy stored in the wood is converted into thermal energy (heat) and light energy.

**Explanation:** A rock at the edge of a cliff has gravitational potential energy due to its elevated position above the ground. This energy can be converted into kinetic energy if the rock falls.

**Explanation:** Kinetic energy is given by \( KE = \frac{1}{2} mv^2 \). If the mass \( m \) is doubled while the velocity \( v \) remains constant, the kinetic energy also doubles.

**Explanation:** The energy stored in food is classified as chemical energy. This energy is released during digestion and used by the body to perform various functions.

**Explanation:** Water at the top of a waterfall has gravitational potential energy. As it flows to the bottom, this potential energy is converted into kinetic energy.

**Explanation:** A stretched bow has elastic potential energy due to its deformation. When released, this energy is converted into kinetic energy as the arrow is launched.

**Explanation:** Radiant energy is the energy transferred by electromagnetic waves, such as light and radio waves. It is a form of energy that can travel through the vacuum of space.

**Explanation:** Kinetic energy is given by \( KE = \frac{1}{2} mv^2 \). If the velocity \( v \) is tripled, the kinetic energy increases by a factor of \( 3^2 = 9 \).

**Explanation:** Thermal energy is the total kinetic and potential energy of the particles in an object. It is related to the temperature of the object and can be transferred through conduction, convection, and radiation.

**Explanation:** Electrical energy is crucial for the operation of electrical appliances as it powers their functions and enables them to perform their tasks.

**Explanation:** Internal combustion engines rely on chemical energy stored in fuels, such as gasoline or diesel, which is converted into mechanical energy to power vehicles.

**Explanation:** The human body metabolizes chemical energy from food, converting it into kinetic energy for movement and thermal energy to maintain body temperature.

**Explanation:** During photosynthesis, plants use radiant energy from sunlight to convert carbon dioxide and water into glucose and oxygen, storing chemical energy.

**Explanation:** In a hydroelectric power plant, the potential energy of stored water is converted into kinetic energy as it flows through turbines, generating electrical energy.

**Explanation:** Wind turbines convert the kinetic energy of moving air (wind) into mechanical energy, which is then transformed into electrical energy.

**Explanation:** Thermal energy from the sun heats the Earth’s surface and atmosphere, driving ocean currents and weather patterns, influencing the climate system.

**Explanation:** Kinetic energy is essential in transportation systems, as it powers the movement of vehicles, such as cars, trains, and airplanes, enabling them to travel from one place to another.

**Explanation:** Geothermal power plants harness thermal energy from the Earth’s interior, using it to generate steam that drives turbines and produces electrical energy.

**Explanation:** Energy conservation is crucial for sustaining energy resources, reducing environmental impact, and ensuring that future generations have access to necessary energy supplies.

**Explanation:** The scalar product, or dot product, of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is defined as \( \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \), where \(\theta\) is the angle between the vectors.

**Explanation:** The dot product \(\mathbf{A} \cdot \mathbf{B} = (2)(4) + (3)(-1) = 8 – 3 = 5\).

**Explanation:** The scalar product of orthogonal (perpendicular) vectors is zero because the cosine of the angle \(90^\circ\) between them is zero.

**Explanation:** If the scalar product of two vectors is zero, it means that the cosine of the angle between them is zero, indicating that the vectors are orthogonal (perpendicular).

**Explanation:** The scalar product of a vector \(\mathbf{A}\) with itself is \(\mathbf{A} \cdot \mathbf{A} = |\mathbf{A}|^2\).

**Explanation:** The dot product of \(\mathbf{A} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{B} = c\mathbf{i} + d\mathbf{j}\) is \(\mathbf{A} \cdot \mathbf{B} = ac + bd\).

**Explanation:** Geometrically, the dot product represents the projection of one vector onto the other, scaled by the magnitude of the second vector.

**Explanation:** Work done by a force is calculated as the dot product of the force vector and the displacement vector: \( W = \mathbf{F} \cdot \mathbf{d} \).

**Explanation:** \(\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} = \frac{23}{\sqrt{5^2 + 2^2} \sqrt{3^2 + 4^2}} = \frac{23}{\sqrt{29} \sqrt{25}} = \frac{23}{\sqrt{725}}\). Solving for \(\theta\) gives approximately \(45^\circ\).

**Explanation:** When two vectors are parallel, the angle between them is \(0^\circ\), and the cosine of \(0^\circ\) is 1. Thus, their scalar product is equal to the product of their magnitudes.

**Explanation:** Geometrically, the scalar product (dot product) of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is equal to \(|\mathbf{A}| |\mathbf{B}| \cos \theta\), where \(\theta\) is the angle between the vectors.

**Explanation:** If the scalar product of two vectors is zero, the cosine of the angle between them is zero, implying that the vectors are perpendicular (orthogonal).

**Explanation:** The dot product of two vectors is given by the formula \(\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta\), where \(\theta\) is the angle between the vectors.

**Explanation:** A positive scalar product indicates that the cosine of the angle between the vectors is positive, meaning the angle is less than \(90^\circ\).

**Explanation:** A negative scalar product occurs when the cosine of the angle between the vectors is negative, meaning the angle is greater than \(90^\circ\).

**Explanation:** When two vectors are aligned in the same direction, the angle between them is \(0^\circ\), and the cosine of \(0^\circ\) is 1, resulting in a positive scalar product.

**Explanation:** The scalar product \(\mathbf{A} \cdot \mathbf{B}\) can be interpreted as the projection of \(\mathbf{A}\) onto \(\mathbf{B}\), multiplied by the magnitude of \(\mathbf{B}\).

**Explanation:** When vectors \(\mathbf{A}\) and \(\mathbf{B}\) are orthogonal, the angle between them is \(90^\circ\), and the cosine of \(90^\circ\) is zero, resulting in a scalar product of zero.

**Explanation:** The scalar product of two vectors is directly related to the cosine of the angle between them, providing a geometric interpretation.

**Explanation:** If one of the vectors is scaled by a factor of 2, the scalar product is also scaled by the same factor, resulting in the product being doubled.

**Explanation:** The scalar product of two vectors is commutative, meaning \(\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}\).

**Explanation:** The self-product property states that the scalar product of a vector with itself is equal to the square of its magnitude: \(\mathbf{A} \cdot \mathbf{A} = |\mathbf{A}|^2\).

**Explanation:** The scalar product of any vector with the zero vector is zero, \(\mathbf{A} \cdot \mathbf{0} = 0\).

**Explanation:** The distributive property of the scalar product states that the scalar product of a vector with the sum of two vectors is equal to the sum of the scalar products: \(\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}\).

**Explanation:** The angle \(180^\circ\) corresponds to \(\cos(180^\circ) = -1\). Thus, the scalar product is negative: \(\mathbf{A} \cdot \mathbf{B} = -|\mathbf{A}| |\mathbf{B}|\).

**Explanation:** The scalar product results in a scalar quantity, which has magnitude but no direction.

**Explanation:** If both vectors are scaled by a factor \(k\), the scalar product is scaled by \(k^2\): \((k\mathbf{A}) \cdot (k\mathbf{B}) = k^2 (\mathbf{A} \cdot \mathbf{B})\).

**Explanation:** The commutative property states that the order of the vectors in the scalar product does not matter: \(\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}\).

**Explanation:** The scalar product of two perpendicular vectors is zero because the cosine of \(90^\circ\) is zero.

**Explanation:** The distributive property allows the scalar product to be distributed over vector addition: \(\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}\).

**Explanation:** The work done by a force is calculated as the scalar product of the force and the displacement, which gives the component of the force in the direction of displacement times the magnitude of the displacement.

**Explanation:** The work done by a constant force is calculated using the scalar product of the force and the displacement: \(W = \mathbf{F} \cdot \mathbf{d}\).

**Explanation:** The work done by the force is given by the scalar product, which involves the cosine of the angle between the force and displacement vectors: \(W = |\mathbf{F}| |\mathbf{d}| \cos \theta\).

**Explanation:** Torque is calculated using the vector product (cross product), not the scalar product. The scalar product is not directly used in the calculation of torque.

**Explanation:** For a variable force, the work done is calculated by integrating the scalar product of the force and the infinitesimal displacement over the path of motion: \(W = \int \mathbf{F}(t) \cdot \mathbf{d}(t) \, dt\).

**Explanation:** In rotational motion, torque is calculated using the cross product of the position vector and the force vector, not the scalar product.

**Explanation:** When the angle between the force vector and the displacement vector is \(90^\circ\), the cosine of \(90^\circ\) is zero, resulting in zero work done: \(W = |\mathbf{F}| |\mathbf{d}| \cos 90^\circ = 0\).

**Explanation:** The work done by a force is calculated using the scalar product, which involves the cosine of the angle between the force and displacement vectors: \(W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| |\mathbf{d}| \cos \theta\).

**Explanation:** Power delivered by a force is calculated as the scalar product of the force and the velocity: \(P = \mathbf{F} \cdot \mathbf{v}\).

**Explanation:** The scalar product of the force and displacement vectors gives the component of the force in the direction of the displacement, which is used to calculate work.

**Explanation:** The work done by a constant force is calculated as the scalar product of the force and displacement vectors: \(W = \mathbf{F} \cdot \mathbf{d}\).

**Explanation:** Since the force and displacement are in the same direction, \(W = F \cdot d = 10 \times 5 = 50\) J.

**Explanation:** The work done by the force is the product of the force, the displacement, and the cosine of the angle between them: \(W = Fd \cos \theta\).

**Explanation:** If there is no displacement, no work is done regardless of the force applied: \(W = F \cdot 0 = 0\).

**Explanation:** The work done by a force perpendicular to the displacement is zero because \(\cos 90^\circ = 0\).

**Explanation:** When the force and displacement are in opposite directions, the work done is negative because \(\cos 180^\circ = -1\).

**Explanation:** \(W = \mathbf{F} \cdot \mathbf{d} = (3\mathbf{i} + 4\mathbf{j}) \cdot (5\mathbf{i} + 2\mathbf{j}) = 3 \times 5 + 4 \times 2 = 15 + 8 = 23\) J.

**Explanation:** The work done is given by \(W = Fd \cos \theta\), showing that it is directly proportional to both the magnitude of the force and the displacement.

**Explanation:** Since the force is applied horizontally and the displacement is also horizontal, \(W = F \cdot d = 10 \times 5 = 50\) J.

**Explanation:** When the force is perpendicular to the displacement, the angle \(\theta = 90^\circ\) and \(\cos 90^\circ = 0\), resulting in zero work being done.

**Explanation:** \(W = \mathbf{F} \cdot \mathbf{d} = 5 \times 3 = 15\) J.

**Explanation:** The horizontal component of the force is \(12 \cos 60^\circ = 12 \times 0.5 = 6\) N. Therefore, \(W = 6 \times 8 = 48\) J.

**Explanation:** \(W = \mathbf{F} \cdot \mathbf{d} = 20 \cos 30^\circ \times 4 = 20 \times \frac{\sqrt{3}}{2} \times 4 = 40\) J.

**Explanation:** The work done \( W \) is calculated using the horizontal component of the force \( F \):
\[ F_{\text{horizontal}} = 15 \cdot \cos 45^\circ = 15 \cdot \frac{\sqrt{2}}{2} = 10.61 \text{ N} \]
Now, calculate the work done:
\[ W = F_{\text{horizontal}} \cdot d = 10.61 \cdot 6 = 63.66 \text{ J} \]

**Explanation:** Since the force and displacement are in the same direction, \(W = F \cdot d = 10 \times 5 = 50\) J.

**Explanation:** The force and displacement are perpendicular (\(\theta = 90^\circ\)), so \(W = F \cdot d \cdot \cos 90^\circ = 15 \times 3 \times 0 = 0\) J.

**Explanation:** The horizontal component of the force is \(8 \cos 30^\circ = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}\) N. Therefore, \(W = 4\sqrt{3} \times 4 = 16\sqrt{3} \approx 27.7\) J.

**Explanation:** The force is applied horizontally, but the displacement is vertically upwards, so \(\theta = 90^\circ\) and \(W = F \cdot d \cdot \cos 90^\circ = 20 \times 5 \times 0 = 0\) J.

**Explanation:** The horizontal component of the force is \(12 \cos 60^\circ = 12 \times 0.5 = 6\) N. Therefore, \(W = 6 \times 8 = 48\) J.

**Explanation:** The horizontal component of the force is \(10 \cos 45^\circ = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2}\) N. Therefore, \(W = 5\sqrt{2} \times 6 = 30\sqrt{2} \approx 42.4\) J.

**Explanation:** The force and displacement are in opposite directions, so the work done is negative: \(W = F \cdot d \cdot \cos 180^\circ = 15 \times 4 \times (-1) = -60\) J.

**Explanation:** The horizontal component of the force is \(25 \cos 60^\circ = 25 \times 0.5 = 12.5\) N. Therefore, \(W = 12.5 \times 10 = 125\) J.

**Explanation:** The force is applied horizontally, but the displacement is vertically upwards, so \(\theta = 90^\circ\) and \(W = F \cdot d \cdot \cos 90^\circ = 30 \times 8 \times 0 = 0\) J.

**Explanation:** The force and displacement are in the same direction, so \(W = F \cdot d \cdot \cos 0^\circ = 18 \times 5 \times 1 = 90\) J.

**Explanation:** The horizontal component of the force is \(40 \cos 30^\circ = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3}\) N. Therefore, \(W = 20\sqrt{3} \times 12 = 240\) J.

**Explanation:** Since the force and displacement are in the same direction, \(W = F \cdot d = 12 \times 8 = 96\) J.

**Explanation:** The force and displacement are in the same direction, so \(W = F \cdot d = 20 \times 5 = 100\) J.

**Explanation:** The horizontal component of the force is \(30 \cos 45^\circ = 30 \times \frac{\sqrt{2}}{2} = 15\sqrt{2}\) N. Therefore, \(W = 15\sqrt{2} \times 10 = 150\sqrt{2} \approx 212.13\) J.

**Explanation:** The force and displacement are in opposite directions, so the work done is negative: \(W = F \cdot d \cdot \cos 180^\circ = 25 \times 6 \times (-1) = -150\) J.

**Explanation:** The horizontal component of the force is \(15 \cos 30^\circ = 15 \times \frac{\sqrt{3}}{2} = 7.5\sqrt{3}\) N. Therefore, \(W = 7.5\sqrt{3} \times 4 = 30\) J.

**Explanation:** The force is applied horizontally, but the displacement is vertically upwards, so \(\theta = 90^\circ\) and \(W = F \cdot d \cdot \cos 90^\circ = 18 \times 6 \times 0 = 0\) J.

**Explanation:** The force is applied at an angle, but the displacement is vertically upwards, so the horizontal component of the force does no work on the object, \(W = F \cdot d \cdot \cos 60^\circ = 40 \times 5 \times 0 = 0\) J.

**Explanation:** Work done by a variable force is defined as the integral of the force over the displacement: \(W = \int \mathbf{F}(x) \cdot d\mathbf{x}\).

**Explanation:** \(W = \int_{1}^{4} (2x + 3) \, dx = \left[ x^2 + 3x \right]_{1}^{4} = (16 + 12) – (1 + 3) = 28 – 4 = 24\) J.

**Explanation:** The SI unit of work is joule (J), which is defined as 1 newton-meter (N·m).

**Explanation:** \(W = \int_{0}^{2} 3x^2 \, dx = \left[ x^3 \right]_{0}^{2} = 8\) J.

**Explanation:** \(W = \int_{1}^{3} \left( 4 + \frac{2}{x} \right) \, dx = \left[ 4x + 2 \ln x \right]_{1}^{3} = (12 + 2 \ln 3) – (4 + 2 \ln 1) = 12 + 2 \ln 3 – 4 = 8 + 2 \ln 3\) J.

**Explanation:** According to the work-energy theorem, the work done by the net force on an object equals the change in its kinetic energy.

**Explanation:** \(W = \int_{1}^{4} 6x \, dx = \left[ 3x^2 \right]_{1}^{4} = 48 – 3 = 45\) J.

**Explanation:** \(W = \int_{0}^{5} 2x \, dx = \left[ x^2 \right]_{0}^{5} = 25\) J.

**Explanation:** \(W = \int_{1}^{3} \frac{6}{x} \, dx = \left[ 6 \ln x \right]_{1}^{3} = 6 \ln 3 – 6 \ln 1 = 6 \ln 3\) J.

**Explanation:** \(W = \int_{0}^{2} 4x^3 \, dx = \left[ x^4 \right]_{0}^{2} = 16\) J.

**Explanation:** \(W = \int_{1}^{4} 5 \, dx = \left[ 5x \right]_{1}^{4} = 20\) J.

**Explanation:** \(W = \int_{1}^{3} (3x^2 + 2x) \, dx = \left[ x^3 + x^2 \right]_{1}^{3} = (27 + 9) – (1 + 1) = 35\) J.

**Explanation:** \(W = \int_{2}^{5} \frac{10}{x} \, dx = \left[ 10 \ln x \right]_{2}^{5} = 10 \ln 5 – 10 \ln 2\) J.

**Explanation:** \(W = \int_{0}^{4} (2x^2 + 3x) \, dx = \left[ \frac{2x^3}{3} + \frac{3x^2}{2} \right]_{0}^{4} = \left( \frac{2 \cdot 64}{3} + \frac{3 \cdot 16}{2} \right) – (0) = \frac{128}{3} + 24 = \frac{128 + 72}{3} = \frac{200}{3} = 48.\overline{3}\) J (approximately 48 J).

**Explanation:** \(W = \int_{1}^{3} 4x \, dx = \left[ 2x^2 \right]_{1}^{3} = 18 – 2 = 16\) J.

**Explanation:** \(W = \int_{0}^{5} 3 \, dx = \left[ 3x \right]_{0}^{5} = 15\) J.

**Explanation:** \(W = \int_{2}^{4} \frac{8}{x} \, dx = \left[ 8 \ln x \right]_{2}^{4} = 8 \ln 4 – 8 \ln 2 = 8 (\ln 4 – \ln 2) = 8 \ln 2\) J.

**Explanation:** Kinetic energy is the energy possessed by an object due to its motion.

**Explanation:** Kinetic energy is calculated using the formula \( KE = \frac{1}{2} mv^2 \), where \(m\) is the mass of the object and \(v\) is its velocity.

**Explanation:** Kinetic energy is directly proportional to the square of velocity, so if velocity doubles, kinetic energy increases by a factor of \(2^2 = 4\).

**Explanation:** The SI unit of kinetic energy (as well as all forms of energy) is the joule (J).

**Explanation:** \( KE = \frac{1}{2} mv^2 = \frac{1}{2} \times 2 \times 3^2 = \frac{1}{2} \times 2 \times 9 = 9 \) J.

**Explanation:** Kinetic energy depends on both the mass (m) and the square of the speed (v) of the object.

**Explanation:** Kinetic energy is proportional to the square of velocity, so if velocity is halved, kinetic energy decreases by a factor of \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \).

**Explanation:** Kinetic energy is the energy associated with the motion of an object.

**Explanation:** \( KE = \frac{1}{2} mv^2 \). Solving for \(v\), \( v = \sqrt{\frac{2 \times KE}{m}} = \sqrt{\frac{2 \times 100}{5}} = \sqrt{40} = 2 \sqrt{10} \approx 4 \) m/s.

**Explanation:** Kinetic energy depends on both the mass and the square of the speed of the object.

**Explanation:** Kinetic energy is proportional to the square of velocity, so if velocity triples, kinetic energy increases by a factor of \(3^2 = 9\).

**Explanation:** According to the work-energy theorem, the work done on an object is equal to its change in kinetic energy.

**Explanation:** \( W = \frac{1}{2} mv^2 = \frac{1}{2} \times 2 \times (10)^2 = 100 \) J.

**Explanation:** Negative work means the force applied is opposite to the direction of motion, resulting in a decrease in kinetic energy.

**Explanation:** First, calculate the work done: \( W = F \cdot d = 5 \cdot 10 = 50 \) J. Then, calculate the initial kinetic energy: \( KE_{\text{initial}} = \frac{1}{2} mv^2 = \frac{1}{2} \times 3 \times (2)^2 = 6 \) J. Finally, the final kinetic energy \( KE_{\text{final}} = KE_{\text{initial}} + W = 6 + 50 = 56 \) J.

**Explanation:** \( W = F \cdot d \cos \theta = 10 \cdot 5 \cdot \cos(0^\circ) = 50 \) J, where \( \theta \) is the angle between the force and the displacement (0 degrees for horizontal motion).

**Explanation:** Work required to bring the object to rest \( W = \frac{1}{2} mv^2 = \frac{1}{2} \times 4 \times (6)^2 = 72 \) J.

**Explanation:** \( W = \frac{1}{2} mv^2 = \frac{1}{2} \times 1000 \times (20)^2 = 20,000 \) J.

**Explanation:** First, calculate the final velocity using \( v^2 = u^2 + 2as \), then use \( W = F \cdot s \).

**Explanation:** To double the kinetic energy, \( KE_{\text{final}} = 2 \times KE_{\text{initial}} = 2 \times 250 = 500 \) J.

**Explanation:** During a car crash, the kinetic energy of the moving vehicle is primarily responsible for causing damage upon impact.

**Explanation:** The kinetic energy of a bullet fired from a gun is due to its motion.

**Explanation:** Wind turbines convert the kinetic energy of moving air (wind) into electrical energy.

**Explanation:** When a hammer strikes a nail, the work done is primarily converted into thermal energy due to friction between the hammer and the nail.

**Explanation:** In a pendulum, kinetic energy is converted into potential energy when it reaches its highest point.

**Explanation:** In a roller coaster ride, potential energy changes significantly as the coaster moves between high and low points.

**Explanation:** As a person jumps from a diving board, potential energy due to height is converted into kinetic energy during the descent.

**Explanation:** When a tennis ball is hit by a racket, potential energy stored in the stretched strings of the racket is converted into kinetic energy of the ball.

**Explanation:** Wind turbines use the kinetic energy of wind to generate electrical energy.

**Explanation:** In a hydroelectric power plant, potential energy of water stored in a reservoir is converted into kinetic energy as it falls through turbines, generating electrical energy.

**Explanation:** Pedaling a bicycle converts chemical energy from the rider’s muscles into kinetic energy, propelling the bicycle forward.

**Explanation:** According to the Work-Energy Theorem, the work done on an object by the net force is equal to the change in its kinetic energy.

**Explanation:** Work done \( W = F \cdot d \cos \theta = 20 \times 5 \times \cos(0^\circ) = 100 \) J, where \( \theta \) is the angle between the force and displacement (0 degrees for horizontal motion).

**Explanation:** The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy.

**Explanation:** If the net work done on an object is zero, its kinetic energy remains constant according to the Work-Energy Theorem.

**Explanation:** Work done \( W = \frac{1}{2} mv^2 \), solve for mass.

**Explanation:** If a force acts on an object causing it to move in the direction of the force, the object’s kinetic energy increases as per the Work-Energy Theorem.

**Explanation:** At the highest point, the ball’s velocity is zero, hence its kinetic energy is zero.

**Explanation:** A compressed spring primarily converts potential energy into kinetic energy when released.

**Explanation:** Negative work implies that the force applied is opposite to the direction of motion, resulting in a decrease in kinetic energy.

**Explanation:** The Work-Energy Theorem relates the work done on an object by the net force to its change in kinetic energy.

**Explanation:** According to the Work-Energy Theorem, the work done by the net force on an object is equal to the change in its kinetic energy.

**Explanation:** Work done \( W = F \cdot d = 10 \times 5 = 50 \) J. Change in kinetic energy \( \Delta KE = W = 50 \) J.

**Explanation:** The Work-Energy Theorem illustrates the principle of conservation of energy in mechanical systems.

**Explanation:** The application of the Work-Energy Theorem relies on Newton’s laws of motion to relate forces and motion to changes in kinetic energy.

**Explanation:** If the net work done on an object is zero according to the Work-Energy Theorem, its kinetic energy remains constant.

**Explanation:** Considering all forces ensures that the total work done on the object accounts for changes in its kinetic energy and potential energy, maintaining conservation of energy.

**Explanation:** Negative work done by a force implies that the force opposes the direction of motion, resulting in a decrease in kinetic energy according to the Work-Energy Theorem.

**Explanation:** The Work-Energy Theorem specifically relates the work done on an object by the net force to its change in kinetic energy.

**Explanation:** The Work-Energy Theorem provides a direct way to relate the work done on an object to changes in its kinetic energy, aiding in the analysis of motion and forces.

**Explanation:** Mechanical energy, in the context of the Work-Energy Theorem, refers to the sum of an object’s kinetic and potential energies.

**Explanation:** Conservation of mechanical energy states that in a system without non-conservative forces (like friction), the total mechanical energy (sum of kinetic and potential energies) remains constant.

**Explanation:** In the absence of friction or other non-conservative forces, a pendulum’s total mechanical energy (kinetic + potential) remains constant due to conservation of mechanical energy.

**Explanation:** As the ball rolls down the hill, potential energy (due to height) decreases and kinetic energy (due to motion) increases, while the total mechanical energy remains constant.

**Explanation:** Conservation of mechanical energy specifically deals with the sum of kinetic and potential energies in a system.

**Explanation:** In an idealized scenario without friction or air resistance, a roller coaster car’s total mechanical energy (kinetic + potential) remains constant as it moves from a higher to a lower position.

**Explanation:** Non-conservative forces like friction convert mechanical energy (kinetic and potential) into other forms such as thermal energy, reducing the total mechanical energy in the system.

**Explanation:** At the highest point, the ball momentarily stops moving upwards and its kinetic energy is zero. Potential energy is at its maximum due to height.

**Explanation:** The law of conservation of energy states that energy cannot be created or destroyed, only transformed. Conservation of mechanical energy is a specific application of this principle.

**Explanation:** As the block slides down, potential energy (due to height) decreases while kinetic energy (due to motion) increases, keeping the total mechanical energy constant.

**Explanation:** Conservation of mechanical energy simplifies the analysis of systems by focusing on the interplay between kinetic and potential energies, especially in scenarios without non-conservative forces.

**Explanation:** Work done \( W = F \cdot d = 5 \times 10 = 50 \) J. Since the force is constant and in the direction of motion, the work done is 20 J.

**Explanation:** According to the Work-Energy Theorem, the work done by the net force on an object equals the change in its kinetic energy. This change in kinetic energy can be used to determine the object’s final speed.

**Explanation:** Potential energy stored in a spring \( PE = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the compression. \( PE = \frac{1}{2} \times 200 \times (0.2)^2 = 2 \) J.

**Explanation:** The Work-Energy Theorem relates the work done by all forces on an object to its change in kinetic energy. In the case of a rocket, the engines do work to accelerate the rocket, thereby changing its kinetic energy and affecting its motion.

**Explanation:** The Work-Energy Theorem directly relates the work done by all forces acting on the skier (like gravity and friction) to changes in the skier’s kinetic energy as they slide down the hill.

**Explanation:** Potential energy, whether gravitational, elastic (like in a spring), or electrical, represents stored energy due to an object’s position or configuration. This energy is crucial in the application of the Work-Energy Theorem.

**Explanation:** According to the Work-Energy Theorem, the work done by the net force on the car equals the change in its kinetic energy. This relationship helps analyze the car’s acceleration over time.

**Explanation:** The Work-Energy Theorem states that the net work done on an object by all forces equals its change in kinetic energy, whether increasing or decreasing.

**Explanation:** For a free-falling object, gravity does work on it, changing its kinetic energy according to the Work-Energy Theorem.

**Explanation:** The Work-Energy Theorem specifically relates the net work done on an object to changes in its kinetic energy, providing a direct relationship between work and energy changes.

**Explanation:** Power is defined as the rate at which work is done or the rate at which energy is transferred or converted.

**Explanation:** Power \( P \) is calculated as the work \( W \) done divided by the time \( t \) taken to do that work.

**Explanation:** \( P = \frac{W}{t} = \frac{5000 \text{ J}}{10 \text{ s}} = 1000 \text{ W} \).

**Explanation:** The SI unit of power is the watt (W), named after James Watt, which is equivalent to one joule per second.

**Explanation:** Power \( P \) can be calculated using \( P = F \times v \), where \( F \) is the force and \( v \) is the velocity. \( P = 200 \text{ N} \times 5 \text{ m/s} = 1000 \text{ W} \).

**Explanation:** Power is a scalar quantity because it has magnitude (amount or size) but does not have direction like vector quantities such as force or velocity.

**Explanation:** Power is calculated as the rate of doing work. The situation with the highest power output would involve lifting the most weight the furthest in the shortest time.

**Explanation:** A generator is designed to convert mechanical energy (often from rotation) into electrical energy efficiently, based on its power output.

**Explanation:** Power is inversely proportional to time \( t \). As \( t \) decreases, \( P = \frac{W}{t} \) increases, assuming work \( W \) remains constant.

**Explanation:** Power represents the rate at which work is done. Higher power means tasks can be completed more quickly because more work is done per unit time.

**Explanation:** The SI unit of power is the watt (W), named after James Watt.

**Explanation:** Horsepower (hp) is a non-SI unit of power commonly used to measure the power output of engines and motors.

**Explanation:** 1 horsepower (hp) is equivalent to approximately 746 watts (W).

**Explanation:** 1 kilowatt (kW) = 1000 watts (W). Therefore, 5000 W = 5 kW.

**Explanation:** Kilowatt (kW) is commonly used to measure the power output of electric appliances due to the practical range of power consumption.

**Explanation:** Energy consumed \( E = P \times t = 60 \text{ W} \times (5 \text{ hours} \times 3600 \text{ s/hour}) = 300000 \text{ J} \).

**Explanation:** 1 horsepower (hp) is approximately equal to 746 watts (W), which equals 0.746 kilowatts (kW).

**Explanation:** Horsepower (hp) is commonly used in contexts involving engines, mechanical work, and gravitational forces.

**Explanation:** Power \( P = \frac{W}{t} = \frac{500 \text{ J}}{10 \text{ s}} = 50 \text{ W} \).

**Explanation:** The watt (W) is named after James Watt, a Scottish engineer who made important contributions to the development of steam engines.

**Explanation:** Power is defined as the rate at which work is done, which is calculated as \( P = \frac{W}{t} \), where \( W \) is the work done and \( t \) is the time taken.

**Explanation:** \( P = \frac{W}{t} = \frac{600 \text{ J}}{3 \text{ s}} = 200 \text{ W} \).

**Explanation:** \( P = \frac{W}{t} = \frac{240000 \text{ J}}{20 \text{ s}} = 12000 \text{ W} \).

**Explanation:** Power is inversely proportional to time \( t \). If \( t \) increases while \( W \) remains constant, \( P = \frac{W}{t} \) decreases.

**Explanation:** Power is calculated as \( P = \frac{W}{t} \). The situation with the most work done in the least time has the highest power output.

**Explanation:** First, calculate power in watts: \( P = \frac{W}{t} = \frac{1200 \text{ J}}{30 \text{ s}} = 40 \text{ W} \). Then convert to kilowatts (kW): \( 0.4 \text{ kW} \).

**Explanation:** Power \( P \) is defined as the amount of work \( W \) done divided by the time \( t \) taken to do that work.

**Explanation:** \( P = \frac{W}{t} = \frac{8000 \text{ J}}{20 \text{ s}} = 400 \text{ W} \).

**Explanation:** First, calculate work \( W = mgh = 50 \text{ kg} \times 9.8 \text{ m/s}^2 \times 10 \text{ m} = 4900 \text{ J} \). Then, calculate power \( P = \frac{W}{t} = \frac{4900 \text{ J}}{5 \text{ s}} = 980 \text{ W} \).

**Explanation:** \( P = \frac{W}{t} = \frac{240 \text{ J}}{12 \text{ s}} = 20 \text{ W} \).

**Explanation:** Potential energy is the energy possessed by an object due to its position or configuration relative to other objects.

**Explanation:** Gravitational potential energy is associated with the position of an object relative to the Earth or another massive body.

**Explanation:** A stretched spring possesses elastic potential energy due to its deformation from its equilibrium position.

**Explanation:** In chemistry, potential energy is often referred to as activation energy, which is the energy required to initiate a chemical reaction.

**Explanation:** A charged particle in an electric field possesses electrical potential energy due to its position in the field.

**Explanation:** A dam stores water at an elevated height, allowing it to possess gravitational potential energy.

**Explanation:** Gravitational potential energy \( E_p = mgh \), where \( m \) is mass, \( g \) is acceleration due to gravity, and \( h \) is height.

**Explanation:** \( E_p \) is elastic potential energy, \( k \) is the spring constant, and \( x \) is the displacement from equilibrium.

**Explanation:** Potential energy refers to stored energy that an object possesses due to its position or configuration.

**Explanation:** The Law of Conservation of Energy states that energy cannot be created or destroyed, only transferred or converted from one form to another.

**Explanation:** Elastic potential energy is the energy stored in a stretched or compressed elastic object, such as a spring.

**Explanation:** \( E_p \) is elastic potential energy, \( k \) is the spring constant, and \( x \) is the displacement from equilibrium.

**Explanation:** \( E_p = \frac{1}{2} kx^2 = \frac{1}{2} \times 50 \text{ N/m} \times (0.2 \text{ m})^2 = 2.0 \text{ J} \).

**Explanation:** Elastic potential energy in a spring depends on its spring constant \( k \) and the displacement from equilibrium \( x \).

**Explanation:** \( E_p = \frac{1}{2} kx^2 \). Doubling \( x \) quadruples \( E_p \), assuming \( k \) remains constant.

**Explanation:** \( E_p = \frac{1}{2} kx^2 \). Changing \( k \) affects force but not \( E_p \) for a given \( x \).

**Explanation:** Elastic potential energy, like other forms of energy, is measured in joules (J).

**Explanation:** A compressed spring stores elastic potential energy due to its deformation from its equilibrium position.

**Explanation:** \( E_p = \frac{1}{2} kx^2 \). Halving \( x \) reduces \( E_p \) to one-fourth, assuming \( k \) remains constant.

**Explanation:** Hooke’s Law relates the force exerted by a spring to its displacement, which is essential for calculating elastic potential energy.

**Explanation:** The principle of conservation of mechanical energy states that in an isolated system, the total mechanical energy (the sum of potential and kinetic energy) remains constant as long as only conservative forces are acting.

**Explanation:** In the absence of non-conservative forces such as friction or air resistance, the total mechanical energy of a system remains constant, according to the principle of conservation of mechanical energy.

**Explanation:** Gravitational force is a conservative force, meaning that the work done by or against it is path-independent and can be fully recovered.

**Explanation:** As the roller coaster car descends, its gravitational potential energy decreases and is converted into kinetic energy, maintaining the total mechanical energy of the system.

**Explanation:** At the lowest point of its swing, the pendulum’s kinetic energy is maximum because its potential energy is at a minimum, and the total mechanical energy is conserved.

**Explanation:** In an isolated system, if the kinetic energy decreases, the potential energy must increase to keep the total mechanical energy constant.

**Explanation:** At the highest point of its trajectory, the projectile’s velocity is zero, so its kinetic energy is zero, and all its initial kinetic energy has been converted into potential energy.

**Explanation:** When only conservative forces are acting on a system, the total mechanical energy remains constant because the energy can be converted between kinetic and potential forms without any loss.

**Explanation:** The work done by a conservative force is path-independent and depends only on the initial and final positions of the object.

**Explanation:** As the block slides down the inclined plane, its gravitational potential energy decreases and is converted into kinetic energy, maintaining the conservation of mechanical energy.

**Explanation:** At the lowest point of its swing, all the pendulum’s potential energy has been converted to kinetic energy, demonstrating the conservation of mechanical energy.

**Explanation:** As the roller coaster car descends, its potential energy decreases and is converted to kinetic energy, keeping the total mechanical energy constant.

**Explanation:** Just before hitting the ground, the block’s potential energy has been completely converted to kinetic energy, which is at its maximum.

**Explanation:** In a perfectly elastic collision, both kinetic energy and mechanical energy are conserved, meaning the total mechanical energy remains constant.

**Explanation:** As the skier descends, their potential energy decreases and is converted to kinetic energy, illustrating the conservation of mechanical energy.

**Explanation:** When the spring is released, its stored potential energy is converted into the kinetic energy of the toy car.

**Explanation:** At the highest point, the ball’s velocity is zero, so its kinetic energy is zero, and its potential energy is at a maximum.

**Explanation:** Assuming no friction, the roller coaster car’s total mechanical energy remains constant as it moves through the loop, converting between kinetic and potential energy.

**Explanation:** At the pendulum’s lowest point, its kinetic energy is at a maximum as all its potential energy has been converted to kinetic energy.

**Explanation:** At the maximum height of the jump, the person’s velocity is zero, so their kinetic energy is zero, and all the energy has been converted to potential energy.

**Explanation:** In the absence of external forces, the total mechanical energy of the satellite remains constant, demonstrating the conservation of mechanical energy in orbital motion.

**Explanation:** The conservation of mechanical energy explains how the car’s potential energy at the top of the hill is converted into kinetic energy as it accelerates down the hill.

**Explanation:** The conservation of mechanical energy allows us to predict that the skier’s potential energy at the top of the slope will be converted into maximum kinetic energy at the bottom, resulting in maximum speed.

**Explanation:** The conservation of mechanical energy is evident throughout the pendulum’s swing as potential energy is converted to kinetic energy and vice versa, maintaining constant total mechanical energy.

**Explanation:** The conservation of mechanical energy principle is utilized, where the potential energy of the stored water is converted into kinetic energy, which is then used to generate electricity.

**Explanation:** For the conservation of mechanical energy to hold, friction and air resistance must be negligible so that there are no non-conservative forces doing work on the system.

**Explanation:** The conservation of mechanical energy explains how the diver’s potential energy at the top is converted to kinetic energy as he falls.

**Explanation:** The conservation of mechanical energy can be used to analyze how the potential energy stored in the compressed spring is converted to the kinetic energy of the ball.

**Explanation:** As the cyclist coasts down the hill, their potential energy is converted into kinetic energy, illustrating the conservation of mechanical energy.

**Explanation:** The conservation of mechanical energy describes how the potential energy stored in the drawn bow is converted into the kinetic energy of the released arrow.

**Explanation:** Hooke’s Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, where F is the force, k is the spring constant, and x is the displacement.

**Explanation:** The spring constant (k) is a measure of the stiffness of the spring. A larger k value indicates a stiffer spring, which requires more force to displace.

**Explanation:** The elastic potential energy stored in a spring when it is compressed or stretched by a displacement x is given by U = 1/2 kx^2, where k is the spring constant.

**Explanation:** Since elastic potential energy is proportional to the square of the displacement (U = 1/2 kx^2), doubling the displacement results in a fourfold increase in potential energy.

**Explanation:** The elastic potential energy is calculated using U = 1/2 kx^2. Substituting k = 200 N/m and x = 0.2 m, we get U = 1/2 * 200 * (0.2)^2 = 4 J.

**Explanation:** The elastic potential energy stored in a spring is proportional to the square of the displacement (U = 1/2 kx^2). Doubling the displacement results in a fourfold increase in potential energy.

**Explanation:** Elastic potential energy is given by U = 1/2 kx^2. Halving the spring constant k while keeping the displacement x constant results in halving the potential energy.

**Explanation:** The work done by a spring when it is stretched or compressed is equal to the elastic potential energy stored in the spring (U = 1/2 kx^2).

**Explanation:** The maximum elastic potential energy is given by U = 1/2 kx^2. Substituting k = 100 N/m and x = 0.3 m, we get U = 1/2 * 100 * (0.3)^2 = 4.5 J.

**Explanation:** In a spring-mass system undergoing simple harmonic motion, the maximum kinetic energy is equal to the maximum elastic potential energy due to the conservation of mechanical energy.

**Explanation:** Using Hooke’s Law, F = kx. Here, F = 100 N and x = 0.5 m. Thus, k = F/x = 100/0.5 = 200 N/m.

**Explanation:** Using Hooke’s Law, F = kx. Here, F = 10 N and x = 0.2 m. Thus, k = F/x = 10/0.2 = 50 N/m.

**Explanation:** Using Hooke’s Law, F = kx. Here, k = 300 N/m and x = 0.1 m. Thus, F = 300 * 0.1 = 30 N.

**Explanation:** Using Hooke’s Law, F = kx. Here, F = 25 N and x = 0.25 m. Thus, k = F/x = 25/0.25 = 100 N/m.

**Explanation:** Using Hooke’s Law, F = kx. Here, F = 15 N and x = 0.3 m. Thus, k = F/x = 15/0.3 = 50 N/m.

**Explanation:** Using Hooke’s Law, F = kx. Here, k = 250 N/m and x = 0.02 m. Thus, F = 250 * 0.02 = 5 N.

**Explanation:** Using Hooke’s Law, F = kx. Here, F = 60 N and x = 0.15 m. Thus, k = F/x = 60/0.15 = 400 N/m.

**Explanation:** Using Hooke’s Law, F = kx. Here, F = 5 N and x = 0.05 m. Thus, k = F/x = 5/0.05 = 100 N/m.

**Explanation:** Using Hooke’s Law, F = kx. Here, k = 150 N/m and x = 0.4 m. Thus, F = 150 * 0.4 = 60 N.

**Explanation:** Using Hooke’s Law, F = kx. Here, k = 80 N/m and x = 0.1 m. Thus, F = 80 * 0.1 = 8 N.

**Explanation:** The potential energy (U) stored in a spring is given by the formula \( U = \frac{1}{2} kx^2 \), where k is the spring constant and x is the displacement from the equilibrium position.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), where k = 200 N/m and x = 0.1 m, the potential energy U = \( \frac{1}{2} \times 200 \times (0.1)^2 = 1 \) Joule.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), where k = 150 N/m and x = 0.2 m, the potential energy U = \( \frac{1}{2} \times 150 \times (0.2)^2 = 3 \) Joules.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), rearrange to find \( k = \frac{2U}{x^2} \). Here, U = 0.125 J and x = 0.05 m, so k = \( \frac{2 \times 0.125}{(0.05)^2} = 100 \) N/m.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), where k = 400 N/m and x = 0.15 m, the potential energy U = \( \frac{1}{2} \times 400 \times (0.15)^2 = 4.50 \) Joules.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), where k = 500 N/m and x = 0.1 m, the potential energy U = \( \frac{1}{2} \times 500 \times (0.1)^2 = 2.5 \) Joules.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), rearrange to find \( k = \frac{2U}{x^2} \). Here, U = 9 J and x = 0.3 m, so k = \( \frac{2 \times 9}{(0.3)^2} = 200 \) N/m.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), where k = 250 N/m and x = 0.2 m, the potential energy U = \( \frac{1}{2} \times 250 \times (0.2)^2 = 5.0 \) Joules.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), rearrange to find \( k = \frac{2U}{x^2} \). Here, U = 0.5 J and x = 0.1 m, so k = \( \frac{2 \times 0.5}{(0.1)^2} = 100 \) N/m.

**Explanation:** Using the formula \( U = \frac{1}{2} kx^2 \), where k = 600 N/m and x = 0.1 m, the potential energy U = \( \frac{1}{2} \times 600 \times (0.1)^2 = 3 \) Joules.

**Explanation:** A pendulum clock uses the potential energy stored in a spring to regulate the movement of the clock’s gears, which in turn keeps accurate time.

**Explanation:** The potential energy stored in a compressed spring is used to launch the ball in a pinball machine.

**Explanation:** The suspension system in vehicles uses springs to absorb shocks from the road, converting kinetic energy into potential energy stored in the springs.

**Explanation:** A toy car with a pull-back motor uses the potential energy stored in a wound-up spring to propel the car forward when released.

**Explanation:** The potential energy stored in a wound spring in a mechanical clock is gradually released to move the clock’s hands, keeping accurate time.

**Explanation:** The potential energy stored in a compressed spring in a jack-in-the-box toy is released to pop the figure out of the box when the lid is opened.

**Explanation:** The springs in a car’s suspension system absorb and dissipate energy from road bumps, providing a smoother ride.

**Explanation:** The potential energy stored in the spring of a mouse trap is released to snap the trap shut when triggered, capturing the mouse.

**Explanation:** Resistance bands use the potential energy stored in stretched springs (or elastic materials) to provide resistance during exercise.

**Explanation:** A pogo stick uses the potential energy stored in a compressed spring to propel the rider upwards when the spring is released.

**Explanation:** Thermal energy is the internal energy of an object due to the kinetic energy of its atoms and molecules. It is associated with the temperature of the object.

**Explanation:** Conduction is the transfer of thermal energy through direct contact between particles in a material, where heat flows from the warmer region to the cooler region.

**Explanation:** Convection occurs in fluids (liquids and gases) when warmer, less dense regions of the fluid rise and cooler, denser regions sink, creating a transfer of heat. Heating water on a stove involves convection currents.

**Explanation:** The joule is the SI unit of energy, including thermal energy. It measures the amount of energy transferred or converted.

**Explanation:** The amount of thermal energy in an object is primarily determined by its temperature, as higher temperatures correspond to greater average kinetic energy of the object’s particles.

**Explanation:** The Law of Conservation of Energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. This includes thermal energy transformations.

**Explanation:** During the phase change from solid to liquid (melting), thermal energy increases the potential energy of the particles, allowing them to move past each other more freely.

**Explanation:** A refrigerator removes thermal energy from its interior and transfers it to the exterior, thus cooling the interior space.

**Explanation:** In a car engine, the thermal energy from the combustion of fuel is converted into mechanical energy to move the vehicle.

**Explanation:** The specific heat capacity is the amount of heat required to raise the temperature of 1 gram of a substance by 1 degree Celsius. It is a measure of how much energy is needed to change the temperature of the substance.

**Explanation:** Chemical energy is the energy stored in the bonds of chemical compounds, such as atoms and molecules. It is released or absorbed during chemical reactions.

**Explanation:** Batteries store chemical energy, which can be converted into electrical energy when the battery is used.

**Explanation:** During a combustion reaction, chemical energy is released as thermal energy (heat) and light energy.

**Explanation:** A fuel cell converts chemical energy from a fuel into electrical energy through a chemical reaction with oxygen or another oxidizing agent.

**Explanation:** In plants, chemical energy is stored primarily in the form of carbohydrates, such as glucose, which are produced during photosynthesis.

**Explanation:** For most living organisms, the primary source of chemical energy is food, which contains nutrients that are broken down to release energy for biological processes.

**Explanation:** In a car engine, chemical energy from fuel is converted into mechanical energy, which is used to move the car.

**Explanation:** During photosynthesis, plants convert solar energy into chemical energy stored in carbohydrates, which can later be used by the plant or other organisms for energy.

**Explanation:** Exothermic reactions release energy, often in the form of heat, and are primarily responsible for the release of chemical energy in cells, such as during cellular respiration.

**Explanation:** Chemical energy is considered a form of potential energy because it is stored in the bonds of chemical compounds and can be released during chemical reactions.

**Explanation:** Nuclear energy is the energy stored in the nucleus of an atom. It can be released through nuclear reactions such as fission and fusion.

**Explanation:** Nuclear fission is the process of splitting the nucleus of an atom into smaller parts, releasing a large amount of nuclear energy.

**Explanation:** Nuclear fusion is the process in which two light atomic nuclei combine to form a heavier nucleus, releasing a significant amount of energy.

**Explanation:** Uranium, particularly Uranium-235, is commonly used as fuel in nuclear fission reactors because its nucleus can be easily split to release energy.

**Explanation:** In the sun, nuclear fusion primarily converts hydrogen nuclei into helium, releasing vast amounts of energy in the form of light and heat.

**Explanation:** Achieving and maintaining the extremely high temperatures and pressures required for nuclear fusion is a significant technical challenge in developing practical fusion energy systems.

**Explanation:** Nuclear fusion produces less long-lived radioactive waste compared to fission, making it a potentially cleaner and safer source of energy.

**Explanation:** A nuclear reactor is a device used to initiate and sustain controlled nuclear fission reactions, which can be used to generate electricity.

**Explanation:** Nuclear energy is commonly used in medical imaging (e.g., PET scans) and treatments (e.g., radiation therapy for cancer).

**Explanation:** Control rods in a nuclear reactor are used to absorb neutrons and regulate the rate of the nuclear reaction, ensuring it remains stable and controlled.

**Explanation:** Electromagnetic energy is energy carried by electromagnetic waves, which include visible light, radio waves, microwaves, infrared radiation, ultraviolet radiation, X-rays, and gamma rays.

**Explanation:** X-rays have shorter wavelengths compared to radio waves, microwaves, and infrared radiation, making them capable of penetrating materials and used in medical imaging.

**Explanation:** Gamma rays have the highest energy in the electromagnetic spectrum, followed by X-rays, ultraviolet radiation, visible light, infrared radiation, microwaves, and radio waves.

**Explanation:** The sun is the primary source of electromagnetic energy on Earth, emitting a wide range of electromagnetic waves across the entire spectrum.

**Explanation:** Infrared radiation from sunlight is primarily responsible for the sensation of warmth, as it heats objects and surfaces it interacts with.

**Explanation:** Electromagnetic waves, specifically radio waves, are used for radio and television broadcasting, as well as for wireless communication such as mobile phones and Wi-Fi.

**Explanation:** Visible light is commonly used in remote sensing and satellite imagery to capture images of Earth’s surface and atmospheric conditions.

**Explanation:** Ultraviolet (UV) radiation from the sun stimulates the production of vitamin D in the skin, which is essential for calcium absorption and bone health.

**Explanation:** X-rays are commonly used in medical imaging (X-ray radiography) to visualize bones and tissues inside the body, helping in diagnosis and treatment planning.

**Explanation:** Solar panels primarily convert sunlight into electricity through the absorption of visible light, which generates an electric current through the photovoltaic effect.

**Explanation:** Energy conversion is the process by which energy changes from one form to another, such as from mechanical energy to electrical energy.

**Explanation:** A battery converts chemical energy stored in its electrodes into electrical energy through electrochemical reactions.

**Explanation:** A hydroelectric dam converts the kinetic energy of flowing water (mechanical energy) into electrical energy through turbines and generators.

**Explanation:** A microwave oven converts electrical energy into electromagnetic waves (microwaves) which heat and cook food by interacting with water molecules.

**Explanation:** A wind turbine converts the kinetic energy of wind (mechanical energy) into electrical energy through the rotation of its blades and a generator.

**Explanation:** A car engine converts chemical energy stored in gasoline (or fuel) into mechanical energy through combustion, which drives the car’s motion.

**Explanation:** A solar-powered calculator converts solar energy (sunlight) into electrical energy through photovoltaic cells, which power the calculator’s functions.

**Explanation:** A geothermal power plant converts heat energy from within the Earth (geothermal energy) into electrical energy through steam turbines and generators.

**Explanation:** A coal-fired power plant converts the chemical energy stored in coal into electrical energy through combustion, steam generation, and turbines.

**Explanation:** A photocopier converts electrical energy into electromagnetic energy (light), which is used to create images on paper through a photosensitive drum.

**Explanation:** Einstein’s equation, E = mc², describes the equivalence of energy (E) and mass (m), showing that mass can be converted into energy and vice versa.

**Explanation:** ‘c’ in the equation E = mc² represents the speed of light in a vacuum, which is approximately 3 × 10⁸ meters per second.

**Explanation:** Energy (E) in the equation E = mc² is typically measured in joules (J), which is the SI unit of energy.

**Explanation:** According to E = mc², if the mass (m) of an object increases, its energy (E) also increases proportionally.

**Explanation:** Einstein’s equation (E = mc²) is fundamental in understanding nuclear reactions, such as nuclear fusion, which powers stars and can be harnessed in nuclear energy production.

**Explanation:** Einstein’s equation (E = mc²) suggests that mass can be converted into energy and vice versa, illustrating the equivalence between mass and energy.

**Explanation:** Einstein’s equation (E = mc²) challenges the classical notion that energy is always conserved separately from mass, indicating that mass and energy can interchange under certain conditions.

**Explanation:** Nuclear fission, such as the splitting of uranium atoms, demonstrates the conversion of a small amount of mass into a large amount of energy, as predicted by Einstein’s equation (E = mc²).

**Explanation:** Mass (m) in Einstein’s equation (E = mc²) is measured in kilograms (kg), which is the SI unit of mass.

**Explanation:** According to Einstein’s equation (E = mc²), the energy released (E) is equal to the mass (m) times the speed of light squared (c²), which is approximately 3 × 10^8 joules per kilogram (J/kg).

**Explanation:** ‘c’ represents the speed of light in a vacuum in Einstein’s equation (E = mc²), which is approximately 3 × 10^8 meters per second.

**Explanation:** The equation E = mc² illustrates that a small amount of mass can be converted into a large amount of energy due to the square of the speed of light, c².

**Explanation:** Fusion reactions in stars, such as the conversion of hydrogen into helium, are governed by the principles of mass-energy equivalence, as described by Einstein’s equation (E = mc²).

**Explanation:** According to Einstein’s equation (E = mc²), when energy is released from a system, the mass of the system decreases, demonstrating the conversion of mass into energy.

**Explanation:** Einstein developed the theory of special relativity to explain the equivalence of mass and energy, encapsulated in the equation E = mc².

**Explanation:** Power is defined as the rate at which work is done or energy is transferred or converted.

**Explanation:** The watt (W) is the SI unit of power, defined as one joule per second (J/s).

**Explanation:** Power (P) is calculated by dividing work (W) by time (t), as expressed by the formula P = W/t.

**Explanation:** Power is higher when the same amount of work is done in less time, as shown in option B.

**Explanation:** High power means being able to do work or achieve results quickly, often associated with speed.

**Explanation:** Power output in engines measures their capability to perform work over time, indicating their performance potential.

**Explanation:** Friction affects the efficiency and power output of mechanical systems by causing energy losses.

**Explanation:** Power defines how quickly energy is transferred or converted from one form to another.

**Explanation:** Electric power describes the rate at which electrical energy is transferred or converted, measured in watts.

**Explanation:** Power depends on velocity because work and energy transfer rates are influenced by how quickly an object moves.

**Explanation:** Mechanical power is defined as the rate at which work is done or energy is transferred in a mechanical system.

**Explanation:** Mechanical power (P) can be calculated by multiplying the force (F) applied to an object by its velocity (v).

**Explanation:** The watt (W) is the SI unit of mechanical power, representing one joule of work done per second.

**Explanation:** An increase in mechanical power indicates that work is being done at a faster rate, often achieved by increasing either force or velocity.

**Explanation:** Machines like steam engines convert heat energy into mechanical power to perform work.

**Explanation:** Power in machines and devices is primarily used to perform mechanical or electrical work.

**Explanation:** An electric drill converts electrical power from the battery or outlet into mechanical power to drive the drill bit.

**Explanation:** Higher power consumption in devices means they consume energy more quickly.

**Explanation:** Power output in diesel engines is commonly measured in horsepower (hp) or kilowatts (kW).

**Explanation:** Internal combustion engines convert the chemical energy from fuel into mechanical power to drive vehicles and machinery.

**Explanation:** Power (P) is calculated as the rate of doing work, which is 500 joules per second (500 J/s = 500 watts).

**Explanation:** 1 horsepower (hp) is approximately equal to 0.746 kilowatts (kW). Therefore, 150 hp ≈ 150 × 0.746 = 111.9 kW, which rounds to 112 kW.

**Explanation:** Energy consumption (E) = Power (P) × Time (t). Therefore, E = 60 watts × 5 hours = 300 watt-hours.

**Explanation:** Energy consumption (E) = Power (P) × Time (t). Therefore, E = 2.5 kW × 2 hours = 5 kWh.

**Explanation:** Energy produced = Power (P) × Time (t). Therefore, Energy produced = 1.5 MW × 24 hours = 36 MWh.