A truck moving on a smooth horizotanal surface with a uniform speed u is carrying stonedust . If a mass `Delta m` of the stone-dust 'leaks' from the truck in a time` Delta t`, the force needed to keep the truck moving at its uniform speed is

Options

A. `u ( Deltam )/( Delta t)`
B. `Delta m ( du )/( dt)`
C. `u ( Delta m)/(Delta t ) + ( Delta m ) ( du)/( dt)`
D. zero

Answer

To solve the problem, we need to determine the force required to keep the truck moving at a uniform speed \( u \) while it is losing mass \( \Delta m \) of stone dust over a time interval \( \Delta t \).

### Step-by-Step Solution:

1. **Understanding the System**:
The truck is moving with a uniform speed \( u \) and is losing mass \( \Delta m \) of stone dust. The key point here is that the truck is moving on a smooth horizontal surface, meaning there are no external forces acting on it (like friction).

2. **Mass Flow Rate**:
The mass flow rate of the stone dust leaking from the truck can be expressed as:
\[ \text{Mass flow rate} = \frac{\Delta m}{\Delta t} \]

3. **Conservation of Momentum**:
According to the principle of conservation of momentum, if there are no external forces acting on the system, the total momentum of the system must remain constant. The momentum \( p \) is given by:
\[ p = m \cdot v \]
where \( m \) is the mass and \( v \) is the velocity.

4. **Effect of Mass Loss**:
As the mass \( m \) decreases due to the leaking stone dust, the momentum of the truck would normally change unless a force is applied to keep the velocity constant. If the mass decreases and no force is applied, the velocity would increase, which contradicts the condition of uniform speed \( u \).

5. **Thrust Force Calculation**:
To maintain the uniform speed \( u \), we need to apply a thrust force. The thrust force \( F_{\text{thrust}} \) can be calculated using the formula:
\[ F_{\text{thrust}} = V_r \cdot \frac{\Delta m}{\Delta t} \]
where \( V_r \) is the relative velocity of the mass leaving the system.

6. **Relative Velocity**:
The relative velocity \( V_r \) of the stone dust with respect to the truck is zero because the stone dust is moving at the same speed \( u \) as the truck at the moment it leaves the truck. Thus:
\[ V_r = u - u = 0 \]

7. **Final Calculation of Thrust Force**:
Substituting \( V_r = 0 \) into the thrust force equation gives:
\[ F_{\text{thrust}} = 0 \cdot \frac{\Delta m}{\Delta t} = 0 \]

8. **Conclusion**:
Since the thrust force needed to keep the truck moving at a uniform speed \( u \) is zero, we conclude that no external force is required to maintain the speed of the truck despite the loss of mass.

### Final Answer:
The force needed to keep the truck moving at its uniform speed \( u \) is **0**.