A man on a rectilinearly moving cart, facing the direction of motion, throws a ball straight up with respect to himself

Options

A. The ball will always return to him.
B. The ball will never return to him.
C. The ball will return to him if the cart moves with constant velocity
D. The ball will fall behind him if the cart moves with some positive acceleration.

Answer

To solve the problem, we need to analyze the motion of the ball thrown by the man on the moving cart. The key points to consider are the relative motion of the ball and the cart, as well as the effects of acceleration.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The cart is moving with a constant velocity \( v_c \) in a straight line.
- The man throws the ball vertically upward with respect to himself. This means that the initial horizontal velocity of the ball \( v_{x} \) is equal to the velocity of the cart, \( v_c \).

2. **Analyze the Motion of the Ball**:
- When the ball is thrown straight up, it has an initial vertical velocity \( v_{y} \) (upward).
- The horizontal component of the ball's velocity remains \( v_c \) throughout its flight because there are no horizontal forces acting on it (assuming air resistance is negligible).

3. **Determine the Motion of the Cart**:
- The cart continues to move with a constant velocity \( v_c \).
- If the cart is moving with constant velocity, it will cover a certain distance while the ball is in the air.

4. **Consider the Time of Flight**:
- The time the ball spends in the air can be calculated using the vertical motion equations. The ball will rise to a maximum height and then fall back down.
- The total time of flight \( t \) can be determined using the formula for vertical motion:
\[ t = \frac{2v_{y}}{g} \]
where \( g \) is the acceleration due to gravity.

5. **Calculate the Distances**:
- During the time \( t \), the cart travels a horizontal distance \( d_{cart} = v_c \cdot t \).
- The ball also travels horizontally at the same speed \( v_c \), so it will also cover the same horizontal distance \( d_{ball} = v_c \cdot t \).

6. **Conclusion on the Return of the Ball**:
- If the cart is moving with constant velocity, the horizontal distance traveled by the ball and the cart will be the same, meaning the ball will return to the man.
- If the cart accelerates positively, the horizontal distance traveled by the cart will be greater than that of the ball, causing the ball to fall behind the cart.

### Final Options Analysis:
- **Option A**: The ball will always return to him. (Incorrect, as it depends on the cart's motion)
- **Option B**: The ball will never return to him. (Incorrect, as it can return if the cart moves at constant velocity)
- **Option C**: The ball will return to him if the cart moves with constant velocity. (Correct)
- **Option D**: The ball will fall behind him if the cart moves with some positive acceleration. (Correct)