Question

A boy is sitting on the topmost berth in a compartment of a train which is about to stop. He then drops an apple aimed at the open hand of his brother who is sitting vertically below his hands at a distance of about 2 m. Which of the following statements is correct?
A) The apple will fall precisely on the hand of his brother.
B) The apple will fall slightly away from the hands of his brother in the direction of the motion of the train.
C) The apple will fall slightly away from the hands of his brother in the direction opposite to the motion of the train.
D) None of the above

Answer 1

Hint:Since the train is about to stop it suffers retardation. The brother who is sitting vertically below the boy inside the train has the same motion as that of the train and therefore will have the same retardation as that of the train as it comes to a stop.
Formula used: The displacement of a body at an instant $t$ moving with an initial velocity $u$ and acceleration $a$ is given by, $s = ut + \dfrac{1}{2}a{t^2}$

Complete step by step answer.
Step 1: Sketch a figure representing the problem at hand.
The below figure represents the boy dropping the apple to his brother sitting vertically below him.

     

Step 2: Express the displacement of the brother and that of the apple.
As the train is stopping its velocity decreases and the train retards. Thus the acceleration of the train will be negative. The brother who is sitting vertically below the boy inside the train has the same motion as that of the train.
The displacement of the brother is then given by, ${s_b} = ut - \dfrac{1}{2}a{t^2}$, where ${s_b}$ is the displacement of the brother, $u$ is his initial velocity and its acceleration is $ - a$ .
When the apple is in the boy’s hand the apple has the same motion as that of the train. But once the boy drops the apple as the train is coming to a stop, the apple will have no retardation in the horizontal direction. Thus the displacement of the apple will be ${s_A} = ut$ .
This implies that the displacement of the apple will be greater than the brother’s displacement.
Therefore, The apple will fall slightly away from the hands of his brother in the direction of the motion of the train.

Option B is correct

Note: This problem can also be explained based on the concept of pseudo force. Here, as the train is retarding, the apple is dropped. The apple will not fall vertically down but it will make an angle with respect to the vertical. This is because a pseudo force acts in a positive direction on the apple making its path slightly parabolic. Hence, the apple will fall slightly away from the hands of his brother in the direction of the motion of the train.