A ball `A` is thrown up vertically with a speed `u` and at the same instant another ball `B` is released from a height `h`. At time `t`, the speed `A` relative to `B` is

Options

A. u
B. u - 2gt
C. `sqrt (u^(2) - 2gh)`
D. u - gt

Explanation

To solve the problem of finding the speed of ball A relative to ball B at time t, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Ball A is thrown upwards with an initial speed \( u \).
- Ball B is released from a height \( h \) at the same time.

2. **Determine the Velocity of Ball A:**
- The velocity of ball A at time \( t \) can be calculated using the equation of motion:
\[ v_A = u - gt \]
- Here, \( g \) is the acceleration due to gravity (acting downwards).

3. **Determine the Velocity of Ball B:**
- Since ball B is released from rest, its initial velocity is 0. The velocity of ball B at time \( t \) is:
\[ v_B = -gt \]
- The negative sign indicates that ball B is moving downwards.

4. **Calculate the Relative Velocity of A with respect to B:**
- The relative velocity of A with respect to B is given by:
\[ v_{AB} = v_A - v_B \]
- Substituting the expressions for \( v_A \) and \( v_B \):
\[ v_{AB} = (u - gt) - (-gt) \]
- This simplifies to:
\[ v_{AB} = u - gt + gt \]
- The \( -gt \) and \( +gt \) cancel each other out:
\[ v_{AB} = u \]

5. **Conclusion:**
- The speed of ball A relative to ball B at time \( t \) is \( u \).

### Final Answer:
The speed of ball A relative to ball B at time \( t \) is \( u \).

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