A trolley filled with sand is moving with a velocity v on a smooth horizontal surface due to inertia. If the sand falls off at the rate of `mu kg//sec`, the velocity of the trolley as a function of time will be best represented by:

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of the trolley filled with sand as the sand falls off. We will use the principle of conservation of momentum to derive the velocity of the trolley as a function of time.

### Step-by-Step Solution:

1. **Initial Conditions**:
- Let the initial mass of the trolley filled with sand be \( M \).
- The initial velocity of the trolley is \( v \).
- The sand falls off at a rate of \( \mu \) kg/s.

2. **Mass of the Trolley Over Time**:
- After time \( t \), the mass of the sand that has fallen off is \( \mu t \).
- Therefore, the mass of the trolley at time \( t \) is given by:
\[ M_t = M - \mu t \]

3. **Conservation of Momentum**:
- According to the conservation of momentum, the total momentum before the sand falls must equal the total momentum after the sand starts falling.
- Initially, the momentum of the system (trolley + sand) is:
\[ P_{\text{initial}} = M \cdot v \]
- After time \( t \), the momentum of the trolley (with mass \( M - \mu t \) moving with velocity \( v' \)) and the falling sand (with mass \( \mu t \) moving with the same velocity \( v' \)) is:
\[ P_{\text{final}} = (M - \mu t) \cdot v' + \mu t \cdot v' \]
- Simplifying this, we get:
\[ P_{\text{final}} = (M - \mu t + \mu t) \cdot v' = M \cdot v' \]

4. **Setting Initial and Final Momentum Equal**:
- By conservation of momentum, we set the initial and final momentum equal:
\[ M \cdot v = M \cdot v' \]
- Since \( M \) is not zero, we can divide both sides by \( M \):
\[ v = v' \]

5. **Conclusion**:
- This shows that the velocity of the trolley remains constant over time, regardless of the sand falling off. Thus, the velocity of the trolley as a function of time is:
\[ v(t) = v \]

### Final Answer:
The velocity of the trolley as a function of time remains constant and is given by:
\[ v(t) = v \]