A boy sitting on the top most berth in the compartment of a train which is just goint to stop on a railway station, drops an apple aiming at the open hand of his brother situated vertically below his hands at distance of about 2m. The apple will fall:-

Options

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 the above

Explanation

To solve the problem, we need to analyze the motion of the apple dropped by the boy in the train. Here’s a step-by-step breakdown:

### Step 1: Understand the Situation
The boy is sitting in a train that is decelerating (retarding) as it approaches a station. He drops an apple aiming for his brother's hand, which is directly below him at a distance of 2 meters.

**Hint:** Consider the effects of the train's deceleration on the motion of the apple.

### Step 2: Analyze the Forces Acting on the Apple
When the boy drops the apple, it is initially moving with the same horizontal velocity as the train. However, as the train is decelerating, the apple will continue to move forward due to inertia (Newton's first law of motion).

**Hint:** Remember that the apple has an initial horizontal velocity equal to that of the train.

### Step 3: Determine the Motion of the Apple
As the apple falls, it will not only fall vertically due to gravity but will also continue to move horizontally in the direction of the train's initial motion. This results in a parabolic trajectory.

**Hint:** Think about how the horizontal and vertical motions are independent of each other.

### Step 4: Calculate the Time of Fall
The time it takes for the apple to fall 2 meters can be calculated using the equation of motion under gravity:
\[ h = \frac{1}{2} g t^2 \]
where \( h = 2 \, \text{m} \) and \( g \approx 9.81 \, \text{m/s}^2 \). Rearranging gives:
\[ t = \sqrt{\frac{2h}{g}} \]

**Hint:** Use the value of \( g \) to find the time of fall.

### Step 5: Calculate the Horizontal Displacement
During the time \( t \) that the apple is falling, it will also move horizontally. The horizontal distance \( d \) can be calculated using:
\[ d = v_{initial} \cdot t \]
where \( v_{initial} \) is the initial horizontal velocity of the apple (same as the train's velocity).

**Hint:** Consider how far the apple travels horizontally during the time it is falling.

### Step 6: Conclusion
Since the apple is moving forward while falling, it will land slightly away from the brother's hand in the direction of the train's motion. Therefore, the apple will not fall directly into the brother's hand.

**Final Answer:** The apple will fall slightly away from the brother's hand in the direction of the motion of the train.