A passenger in a train drops a ball from the window of a train running at an acceleration 'a'. A pedestrain, on the ground, by the side of the rails, observers the ball falling along

Options

A. a parabola with an acceleration 'g'
B. a parabola with an acceleration `sqrt(g^(2)+a^(2))`
C. a vertical with an acceleration `sqrt(g^(2)+a^(2))`
D. a vertical with an acceleration `sqrt(g^(2)-a^(2))`

Explanation

To solve the problem, we need to analyze the motion of the ball dropped from the train in the context of both the passenger in the train and the pedestrian on the ground.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- The train is moving with an acceleration 'a'.
- A passenger drops a ball from the window of the train.
- We need to analyze the motion of the ball as observed by a pedestrian standing on the ground.

2. **Initial Conditions**:
- When the ball is dropped, it has the same horizontal velocity as the train at that moment. Let's denote this horizontal velocity as \( v_0 \).
- The vertical motion of the ball will be affected by gravity, while the horizontal motion will continue with the initial velocity \( v_0 \).

3. **Horizontal Motion**:
- The horizontal motion of the ball can be described by the equation:
\[ x(t) = v_0 t \]
- Here, \( x(t) \) is the horizontal displacement of the ball from the point it was dropped, and \( t \) is the time after the ball is dropped.

4. **Vertical Motion**:
- The vertical motion of the ball is influenced by gravity. The equation for vertical motion is:
\[ y(t) = \frac{1}{2} g t^2 \]
- Here, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).

5. **Relative Motion**:
- The pedestrian will observe the ball falling at a certain angle because while it is falling vertically, it is also moving horizontally with the initial velocity of the train.
- The trajectory of the ball will be a parabola as seen by the pedestrian.

6. **Final Observations**:
- The pedestrian will see the ball falling along a curved path rather than straight down because of the combination of horizontal motion (due to the train's velocity) and vertical motion (due to gravity).

### Conclusion:
The pedestrian will observe the ball falling along a parabolic path due to the initial horizontal velocity imparted by the train's motion and the vertical acceleration due to gravity.