A truck of mass `500kg` is moving with constant speed `10ms^(-1)`. If sand is dropped into the truck at the constant rate `10kg/min`,the force required to mainatain the motion with constant velocity is .

Options

A. `(3)/(2)N`
B. `(5)/(4)N`
C. `(7)/(5)N`
D. `(5)/(3)N`

Explanation

To solve the problem, we need to determine the force required to maintain the motion of a truck that is gaining mass due to sand being dropped into it. The truck is moving at a constant speed, which implies that the net force acting on it must be zero according to Newton's first law of motion.

### Step-by-step solution:

1. **Identify the given values:**
- Mass of the truck, \( m = 500 \, \text{kg} \)
- Constant speed of the truck, \( v = 10 \, \text{m/s} \)
- Rate at which sand is dropped into the truck, \( \frac{dm}{dt} = 10 \, \text{kg/min} \)

2. **Convert the rate of mass increase into SI units:**
- Since \( 1 \, \text{min} = 60 \, \text{s} \), we convert \( 10 \, \text{kg/min} \) to \( \text{kg/s} \):
\[ \frac{dm}{dt} = \frac{10 \, \text{kg}}{1 \, \text{min}} = \frac{10 \, \text{kg}}{60 \, \text{s}} = \frac{1}{6} \, \text{kg/s} \]

3. **Use the formula for force related to momentum change:**
- The force required to maintain constant velocity while the mass of the truck is changing can be calculated using the formula:
\[ F = v \cdot \frac{dm}{dt} \]
where \( F \) is the force, \( v \) is the velocity, and \( \frac{dm}{dt} \) is the rate of change of mass.

4. **Substitute the known values into the formula:**
\[ F = 10 \, \text{m/s} \cdot \frac{1}{6} \, \text{kg/s} \]
\[ F = \frac{10}{6} \, \text{N} = \frac{5}{3} \, \text{N} \]

5. **Final answer:**
- The force required to maintain the motion of the truck with constant velocity is:
\[ F = \frac{5}{3} \, \text{N} \approx 1.67 \, \text{N} \]