Ball A is dropped from rest from a window. At the same instant, ball B is thrown downward, and ball C is thrown upward from the same window. Which statement concerning the balls after their release is necessarily true if air resistance is neglected ?

Options

A. At some instant after it is thrown, the acceleration of ball C is zero.
B. All three balls strike the ground at the same time.
C. All three balls have the same velocity at any instant.
D. All three balls have the same acceleration at any instant.

Answer

To solve the problem, we need to analyze the motion of three balls (A, B, and C) released from the same height but with different initial conditions.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Ball A is dropped from rest, so its initial velocity \( u_A = 0 \).
- Ball B is thrown downward, so it has some initial downward velocity \( u_B \) (which is greater than 0).
- Ball C is thrown upward, so it has some initial upward velocity \( u_C \) (which is also greater than 0).

2. **Identifying the Forces**:
- All three balls are subjected to the force of gravity, which causes them to accelerate downwards at a constant rate \( g \) (approximately \( 9.81 \, \text{m/s}^2 \)).
- The acceleration for all three balls is \( a = g \) downward.

3. **Using the Equations of Motion**:
- For each ball, we can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- Where \( s \) is the distance fallen (which is the same for all three balls, equal to the height of the window), \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time of flight.

4. **Comparing the Time of Flight**:
- For Ball A:
\[ H = 0 \cdot t_A + \frac{1}{2} g t_A^2 \implies H = \frac{1}{2} g t_A^2 \]
- For Ball B:
\[ H = u_B t_B + \frac{1}{2} g t_B^2 \]
- For Ball C:
\[ H = u_C t_C - \frac{1}{2} g t_C^2 \]
- Since the initial velocities \( u_B \) and \( u_C \) are different from \( u_A \), the times \( t_A \), \( t_B \), and \( t_C \) will also be different.

5. **Conclusion on Acceleration**:
- Despite the differences in initial velocities and times of flight, the acceleration of all three balls is the same at any instant because they are all in free fall under the influence of gravity. Therefore, the statement that "all three balls have the same acceleration at any instant" is true.

### Final Answer:
The correct statement concerning the balls after their release is that **all three balls have the same acceleration at any instant**.

---