A boy sitting on the topmost berth in the compartment of a train which is just going to stop on railway station, drops an apple aiming at the open hand of his brother sitting vertically below his hands at a distance of about 2 meter. The apple will fall

Options

A. Prescisely on the hand of his brother.
B. Slightly away from the hand of his brother in the direction of motion of the train
C. slightly away from the hand of his brother opposite to the direction of motion of the train
D. none of the above

Answer

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 solution:

### Step 1: Understand the Scenario
The boy is sitting in a train that is moving horizontally. He drops an apple aiming at his brother who is sitting directly below him, at a distance of 2 meters.

**Hint:** Visualize the situation by drawing a diagram of the train, the boy, and his brother.

### Step 2: Identify the Initial Conditions
When the boy drops the apple, it has an initial horizontal velocity equal to the velocity of the train (let's denote this as 'u'). The apple is dropped from rest in the vertical direction, meaning its initial vertical velocity is 0 m/s.

**Hint:** Remember that the apple's horizontal motion is independent of its vertical motion.

### Step 3: Analyze the Vertical Motion
The apple will fall under the influence of gravity. The time it takes to fall 2 meters can be calculated using the equation of motion:
\[ y = \frac{1}{2} g t^2 \]
where:
- \( y \) = 2 m (the vertical distance)
- \( g \) = 9.81 m/s² (acceleration due to gravity)
- \( t \) = time in seconds

Rearranging the equation to solve for time \( t \):
\[ t = \sqrt{\frac{2y}{g}} \]

**Hint:** Substitute the values of \( y \) and \( g \) to find the time taken for the apple to fall.

### Step 4: Calculate the Horizontal Distance
During the time \( t \) it takes for the apple to fall, it will also continue to move horizontally with the same speed as the train. The horizontal distance \( x \) covered by the apple can be calculated using:
\[ x = u \cdot t \]

**Hint:** Use the value of \( t \) you calculated in the previous step to find the horizontal distance.

### Step 5: Conclusion
Since the apple has a horizontal velocity (equal to the speed of the train) while it is falling, it will land at a point that is horizontally displaced from the brother's hand. Therefore, the apple will fall slightly away from the brother's hand in the direction of the train's motion.

**Hint:** Conclude by stating that the apple does not land directly in the brother's hand due to the horizontal motion.

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