1. There is no force acting on a moving body. What will be state of motion for the body finally?

The body continues to move with same uniform velocity (i.e. along a straight line). This follows from the 1st or 2nd law of Newton.

2. A girl sitting in a railway carriage, moving at uniform velocity, throws a ball straight up in the air. Where will the ball fall?

The ball will fall into her hand in the absence of any air-resistance.

This can be explained from the inertia of motion of the ball. Before the projection the ball was moving in the horizontal direction along with the carriage. At the moment of projection the vertical velocity is added. The combined velocity takes the ball in a parabolic path. During its motion the vertical component is affected by the graity but the horizontal motion (due to inertia) remains uniform. Thus the ball covers the same horizontal distance as the girl during the time of flight of the ball.

3. In the previous question, if the train slows down while the ball is in air – where would the ball fall?

The ball falls in front of the girl.

While the ball is in air it travels with a horizontal velocity that remains same as the velocity of the carriage at the time of projection. So, if the carriage slows down it cannot travel the same horizontal distance as the ball. Therefore the horizontal distance traveled by the ball becomes larger than that of the carriage and falls ahead of the girl.

4. While you are driving a car at a constant speed along a straight path, you see a straight wall in front of you at a distance . The wall is oriented perpendicular to your direction of motion. Will you apply a brake or take turn without reducing the speed to avoid the collision, taking into account the force required for these actions?

Applying brake is the better option as the force required is smaller.

The force () required to stop the car before hitting the wall:

Using the formula

   

we can get acceleration

   

Therefore, the required force

   

The force () required to turn the car before hitting the wall:

Here the car travels along a circular path of radius to avoid the collision. Therefore, the required force is a centripetal force.

And the required force

   

Comparing the two forces we see that that is less force is required when the brake is applied.

5. When a ball is thrown in the upward direction, first its momentum decreases and then it increases again during its fall. Does it violate the principle of conservation of momentum?

The principle of conservation of momentum (of a body) is valid when there is no external force on the body. As the ball is acted on by the force of gravity of earth – its momentum cannot remain constant during its motion. If we consider the ball and the earth forming one single system then the force of gravity cannot be treated as an external force and the total momentum of this system remains conserved. That means the change of momentum of the ball becomes equal and opposite to the change of momentum of the earth at any time during the time the ball is in motion.

6. Two moving bodies of different masses have same momentum. Which one has greater kinetic energy?

The lighter body has greater kinetic energy.

Let the masses of the two bodies are and (with ). The velocities be and , respectively. As given –

   

and therefore

   

.

Now comparing the kinetic energies ( and ) we get

   

Hence

   

or the lighter body has the greater kinetic energy.

7. Two moving bodies of different masses have same kinetic energy. Which one has greater momentum?

The heavier body has greater momentum.

Let the masses of the two bodies are and (with ). The velocities be and , respectively. As given –

   

and therefore

   

Now comparing the momenta ( and ) we get

   

Hence

   

or the heavier body has the greater momentum.

8. Why a man hits harder when he falls on a paved floor than when he falls on the sand from same height?

Let be the height from which the man (of mass ) falls. So, the kinetic energy of the man just before hitting the ground is same as his potential energy () at height . This kinetic energy is used to do the work against the resistance of the paved floor or sand before he stops. Let the resistance (upward) force be and he moves through a distance before he stops. Equating the work done by the net force to stop the man and the kinetic energy at the time of hitting the ground we get,

   

or,

   

So, the resisting force is bigger if is smaller and vice versa. For a paved surface is much smaller than that in case of sand.

So, the resistance by the paved floor is much bigger compared to that for sand and thus hits harder.

9. If the earth shrinks to half its radius keeping its mass same how would be the change in the length of a day?

This can be obtained from the conservation of angular momentum of the earth. So, we get

   

or,

   

Here and are the moments of inertia of the earth before and after the change. and are the corresponding angular speeds of the earth. So, the earth rotates 4 times faster after it shrinks. Therefore, the length of a day is in this new situation.

10. Two bodies of same mass, moving at same speed toward each other, collide and stick together. How will be their motions after the collision?

As the masses and the speeds are same, so the momenta of the two bodies are equal in magnitude. But they travel in the opposite directions. Therefore, the momenta of the two bodies are equal and opposite producing a net zero momentum of the system. This collision is inelastic and after the collision, a combined mass having zero momentum is there. This implies that the combined mass is at rest after the collision.

11. A body travels on a frictionless plane at a certain speed and hits a second body of same mass staying at rest on the plane. Describe their motions after the impact.

In case of a one-dimensional collision between tow bodies of same mass, the velocities are exchanged. So, the first body stops after the collision while the second body starts to move at same velocity as the first one.

12. A diver curls her body when she dives – why?

In the process of curling, the diver pulls in her limbs close to the center of the body. This reduces the moment of inertia () of the body about the body-centre. At the moment of diving, the diver develops a torque to produce the rotation in the body. During the fall no more torques act on the body keeping the total angular momentum (ω) of the body conserved. Therefore the reduction of moment of inertia () causes the increase of angular velocity (ω) and the diver can complete bigger number of rotations during her fall.