Question

A boy sitting on the top most berth in the compartment of a train which is just going to stop at a railway station, drops an apple aiming at the open hand of his brother situated vertically below his hands at a distance of about 2m. The apple will fall
A. In the hand of his brother
B. Slightly away from the hands of his brother in the direction of motion of the train.
C. Slightly away from the hands of his brother in the direction opposite to the direction of motion of the train.
D. None of these.

Answer 1

Hint: For the apple to fall in the hand of the boy, the apple and the boy’s hand must cover the same horizontal distance after the apple is released. The apple will have its horizontal speed equal to the speed of the train at the time of release. Calculate the distance with d=vt. Then check whether the boy's hand covers the same distance.

Formula used:
d = vt

Complete answer:
Let us understand the given situation. There are two boys (bothers) sitting in a moving train. One boy is sitting exactly at the top of the other boy at a height of 2 metres.
The train is about to stop at a station. This means that the train is under retardation. In other words, the train is accelerating in the opposite direction to the direction of its velocity.
The boy sitting at the top drops an apple aiming at the hand of his brother sitting down. Let the speed of the train at this instant be v. Since the boy is sitting inside the train, he will also be moving with the same speed v. The apple is in contact with the boy. Therefore, when the apple is released, its velocity will be v along the direction of the motion of the train.

After this instant, the apple will come down under the influence of the gravitational pull exerted by earth. However, the direction of gravitational force is downwards. Therefore, the apple will accelerate downwards but it will not accelerate in the horizontal direction since there is no force acting on it in that direction. Therefore, the apple will move with constant velocity v in the horizontal direction.
Let the time taken for the apple to come down from a height of 2m be t. In this time ‘t’ the apple will cover a distance equal to d = vt.
Therefore, for the apple to fall in the hands of the second boy, the hand of the boy must also cover a horizontal distance of vt. However, the train is under retardation, which means that its velocity is decreasing. This means that the distance that the boy’s hand will cover in time t is less than vt.
Since the boy’s hand covers less distance than that of the apple, the apple will fall away from this hand in the direction of the motion of the train.
Hence, the correct option is B.

Note: We can also analyse the case with respect of the train. With respect to the train, both boys are stationary.
When the apple is released, a pseudo force will act on the apple in the direction opposite to the direction of the acceleration of the train. Suppose the train is moving along the direction of the positive x-axis. Then its acceleration will be in the direction of the negative x-axis. Hence, the pseudo force will be in the direction along the positive x-axis (i.e. along the motion of the train).
Therefore, with the train the boy is stationary and the apple is accelerated horizontally. Hence, the apple will not fall in the boy’s hand.