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. `2u`
C. `u-g t`
D. `sqrt(u^(2)-g t)`

Explanation

To find the speed of ball A relative to ball B at time \( t \), we can follow these steps:

### Step 1: Determine the speed of ball A after time \( t \)
Ball A is thrown upwards with an initial speed \( u \). As it moves upwards, it experiences a downward acceleration due to gravity \( g \). The speed of ball A after time \( t \) can be calculated using the equation of motion:
\[ V_A = u - gt \]
where:
- \( V_A \) is the speed of ball A after time \( t \),
- \( u \) is the initial speed of ball A,
- \( g \) is the acceleration due to gravity,
- \( t \) is the time elapsed.

### Step 2: Determine the speed of ball B after time \( t \)
Ball B is released from a height \( h \) and starts from rest. It falls under the influence of gravity, so its speed after time \( t \) can be calculated as:
\[ V_B = gt \]
where:
- \( V_B \) is the speed of ball B after time \( t \).

### Step 3: Calculate the relative speed of ball A with respect to ball B
The relative speed of ball A with respect to ball B is given by:
\[ V_{A \text{ relative to } B} = V_A - V_B \]
Substituting the expressions for \( V_A \) and \( V_B \):
\[ V_{A \text{ relative to } B} = (u - gt) - (gt) \]
This simplifies to:
\[ V_{A \text{ relative to } B} = u - gt + gt = u \]

### Conclusion
The speed of ball A relative to ball B at time \( t \) is:
\[ \boxed{u} \]