1. (a) True.
2. (b) True.

1. Yes, mechanical energy comprises both potential energy and kinetic energy.
2. Zero momentum means that velocity is zero. Hence, there is no kinetic energy, but the object may still possess potential energy.

1. Power is defined as the rate at which work is done or energy is transferred. It tells us how quickly a task is completed.

1. Power is equal to the total work done divided by the time taken to do that work.

1. Mass of the cricket ball = 0.2 kg. Velocity = 154.8 km/h ≈ 43 m/s.

1. K = ½ × 0.2 × (43)² = 184.9 J.

1. Commercial unit of energy = kilowatt hour (kWh). SI unit of energy = Joule (J).
2. 1 kWh = 1 kW × 1 h = 1000 W × 3600 s = 1000 Js⁻¹ × 3600 s = 3,600,000 J = 3.6 × 10⁶ J.

1. Where K.E. is the kinetic energy of an object, 'm' is the mass of the object, and 'v' is its velocity.

1. All cutting instruments such as blades, axes etc. are sharpened so that the area of cross-section decreases, and hence the pressure exerted by them increases.
2. Thus they can easily penetrate a given surface.

1. mgh = 0.2 kg × 10 m s⁻² × 10 m = 20 J.

1. In the case of a body moving along a circular path, the centripetal force is always along the radius, while displacement is tangential.
2. Hence, work done W = Fs cos90° = 0, as the angle between F and s is 90°.

1. When a person holds a bundle of hay over his head, he applies a force in the upward direction, but there is no displacement of the bundle (it's held stationary).
2. W = F × s cosθ = F × 0 = 0. No work is done, even though he feels tired.

1. Yes, it is possible. This occurs if an object is moving in a circular path, because the force (centripetal) is always acting perpendicular to the direction of displacement (tangential).
2. Since W = Fs cos90° = 0, no work is done even though the object is accelerating (changing direction).

1. Given: F = 10 N, s = 2 m, θ = 60°.
2. W = 10 × 2 × cos60° = 10 × 2 × ½ = 10 J.

1. (a) Positive
2. (b) Friction
3. (c) Displacement

1. Let height of each floor = h. Potential energy PE = mgh.
2. For 10th floor → height = 10h. E₁ = mg(10h) = 10mgh.
3. For 20th floor → height = 20h. E₂ = mg(20h) = 20mgh. So E₂ = 2E₁.

Power required

1. Let time for 10th floor = t, time for 20th floor = 2t.
2. Power for 10th floor: P₁ = 10mgh/t. Power for 20th floor: P₂ = 20mgh/2t = 10mgh/t.
3. So, P₂ = P₁ (power required stays the same).

1. According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy.
2. When positive work is done, the object's kinetic energy increases because the force acts in the direction of motion.
3. When negative work is done, the kinetic energy decreases as the force opposes motion.
4. If no work is done, there is no change in kinetic energy, and the object continues moving with the same speed.

1. (i) PE = mgh = 50 × 10 × 72.5 = 36,250 J.
2. (ii) PE = mgh = 50 × 10 × 72.5 = 36,250 J.
3. (iii) The gain in potential energy is the same in both cases, showing that potential energy depends only on the initial and final positions (height), not on the path taken to reach there.

1. All real machines slow down and stop because external forces like friction and air resistance do negative work on them.
2. This means the machine's mechanical energy decreases, and energy is continuously converted into heat and sound.
3. No machine can perfectly conserve its mechanical energy, since energy is being lost to the surroundings, so the machine gradually loses motion and finally stops.

1. In science, work is not defined by effort or tiredness, but by force causing displacement.
2. A person may feel tired due to energy used by muscles, but if there is no movement, no work is done scientifically.
3. Example: When pushing a rigid wall, the person uses energy and feels tired, but the wall does not move. Since displacement is zero, the work done on the wall is also zero, showing the difference between effort and scientific work.

1. Let the aircraft's approach velocity be v. Initial KE = ½ × 15000 × v². Final KE = 0.
2. The displacement of the aircraft and the force applied by the wire are in opposite directions, so the work done by the wire is negative: W = F × s = 367500 N × (−100 m).
3. Using work done = change in KE: −½ × 15000 × v² = −(367500 × 100).
4. v² = (367500 × 2 × 100)/15000 = 4900. v = √4900 = 70 m/s (≈ 252 km/h) towards the aircraft carrier.

1. (i) Kinetic energy of belt/package, heat energy, and sound energy.
2. (ii) m = 36 kg, h = 2.4 m, g = 10 m/s². G.P.E. = m × g × h = 36 × 10 × 2.4 = 864 J.
3. (iii) Power = W/t = 864/4.4 = 196.36 W.
4. (iv) Since power is constant and mass is increased, the increase in potential energy is greater. Also, as mass increases, speed reduces, and hence the time taken by the conveyor is longer.

1. When the child is at the top of the slide, they have gravitational potential energy due to height.
2. As the child slides down, potential energy decreases and kinetic energy increases.
3. At the bottom, almost all the potential energy converts into kinetic energy (assuming negligible friction). Applying conservation of mechanical energy:

1. Mass cancels out, giving: v = √2gh, where h = height of slide and g = acceleration due to gravity.

1. Scissors work as a lever with a fulcrum (pivot point).
2. When the object is placed closer to the fulcrum, the effort arm becomes longer and the load arm becomes shorter.
3. This increases the mechanical advantage, making it easier to cut hard objects.

1. An object gains the capacity to do work when energy is transferred to it by doing work on it.
2. When a force acts on an object and causes displacement, work is done, and the object gains energy, usually as kinetic or potential energy.
3. This stored energy enables the object to apply force on another object and cause motion. For example, a moving ball can hit another object and transfer its energy, showing its ability to do work.

1. (i) Potential Energy is the energy possessed by a body by virtue of its position or shape.
2. (ii) In a toy car, the wound spring possesses potential energy. When the spring is released, its potential energy changes into kinetic energy, causing the toy car to move.
3. (iii) Given: m = 50 kg, h₁ = 75 m, h₂ = 60 m. At A: PE₁ = mgh₁ = 50×10×75 = 37,500 J. At B: PE₂ = mgh₂ = 50×10×60 = 30,000 J.
4. Change in PE = PE₁ − PE₂ = 37500 − 30000 = 7500 J.

OR

1. The kinetic energy of an object of mass m moving with velocity v is given by ½mv².
2. To bring the object to rest, an equal amount of work, i.e. ½mv², is required to be done on the object (as negative work).

1. (i) Mechanical energy is the total energy of an object due to its motion and position. It is calculated as the sum of kinetic energy and potential energy present in the system.
2. (ii) The two components of mechanical energy are kinetic energy and potential energy. Kinetic energy depends on motion, while potential energy depends on position, especially height in gravitational fields.
3. (iii) In a falling object, potential energy decreases as height reduces, while kinetic energy increases due to rising speed. The total mechanical energy remains constant because the loss in potential energy equals the gain in kinetic energy.

OR

1. Conservation of mechanical energy simplifies problem-solving by avoiding detailed force calculations. It allows us to determine velocity or position by considering energy changes, making it easier to analyse motion in many real-life and physics problems.

1. (i) Law of conservation of energy: The energy in a system can neither be created nor destroyed. It may be transformed from one form to another, but the total energy of the system remains constant.
2. (ii) The energy possessed by a body due to a displacement caused in it by the application of a force is called mechanical energy.
3. (iii) Energy = Power × Time = 4 × 600 W × 5 h = (4×600×5)/1000 = 12000/1000 = 12 kWh.

OR

1. Yes, a body can have energy without possessing momentum. A body may have potential energy and still be at rest, meaning it would have zero momentum and yet still possess energy.

1. (a) Negative
2. (b) Displacement
3. (c) 1 joule
4. (d) ½mv²
5. (e) mgh

(a) Examples

1. (i) A cricket/hockey ball having small mass which has been hit hard and is travelling fast (possesses high kinetic energy).
2. (ii) A shot put having large mass thrown by an athlete possesses low kinetic energy.

(b) Proof of conservation of mechanical energy

1. Let a body of mass m be at height h above the ground at point A. Its K.E. = 0, P.E. = mgh. Total energy at A = 0 + mgh = mgh.
2. After falling freely through a distance x, at point B: KE at B = mgx, and PE = mg(h−x). Total energy at B = mgx + mg(h−x) = mgh.
3. When it reaches the ground at point C: v² = u² + 2gh, and since u=0, v² = 2gh. Its KE = ½mv² = ½m(2gh) = mgh, and PE = 0. Total energy at C = mgh + 0 = mgh.

1. (i) Energy can neither be created nor destroyed; it may be transformed from one form to another, but the total energy of an isolated system remains constant.

(ii) Pendulum oscillation

1. Consider a simple pendulum in its mean position P. When pulled to position R, it gains height, so at R it has: maximum potential energy, and zero kinetic energy (held by hand).
2. When released from R, it moves towards P and velocity increases, so kinetic energy increases at the expense of potential energy. At position P (lowest point), it has: maximum kinetic energy, and zero potential energy.
3. When it swings from P to Q, it gains height again, so PE increases while velocity (and hence KE) decreases. At Q it has: maximum potential energy, and zero kinetic energy momentarily.
4. From this example, it is clear that in the pendulum-Earth system, energy is conserved — it is the potential energy which changes into kinetic energy and vice-versa.

1. When an object is raised through a certain height above the ground, its energy increases because work is done on it against gravity.
2. The energy present in such an object is called gravitational potential energy — the work done in raising it from the ground to that point against gravity.
3. Consider a body of mass m at point P on the Earth's surface (PE = 0). To lift it to point Q at height h, we apply a minimum force equal to mg (upward), doing work against gravity.

1. As F = mg and s = h, substituting: W = mg × h = mgh.
2. This work done equals the gain in energy of the body — the potential energy.

1. (i) Energy can be transferred without direct contact through radiation. For example, the Sun transfers energy to the Earth in the form of heat and light through space, without any physical contact between them.
2. (ii) Energy can also be transferred as heat. When two objects at different temperatures come into contact, energy flows from the hotter object to the colder one. This process continues until both objects reach the same temperature, known as thermal equilibrium, showing transfer of energy without mechanical work.

1. When the pendulum bob is pulled (say towards left), the energy supplied is stored as P.E. due to its higher position.
2. When released and moving towards right, its P.E. changes into K.E., such that at the mean position it has maximum K.E. and zero P.E.
3. As the pendulum moves towards the extreme right, its K.E. changes back into P.E., reaching zero K.E. at the extreme. Moving from this extreme back to mean position, P.E. again changes to K.E. This illustrates the law of conservation of energy.

Why does the bob eventually stop?

1. Eventually, the bob comes to rest because during each oscillation, a part of its energy is transferred to air (air resistance) and lost in overcoming friction at the point of suspension.
2. Thus, the energy of the pendulum is dissipated into the surroundings (as heat/sound in air and at the pivot).