A cart is moving with a velocity 20m/s. Sand is being dropped into the cart at the rate of 50 kg/min. The force required to move the cart with constant velocity will be :-

Options

A. 50 N
B. 30.33N
C. 26.45N
D. 16.66N

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
We have a cart moving with a constant velocity of 20 m/s, and sand is being dropped into the cart at a rate of 50 kg/min. We need to find the force required to keep the cart moving with this constant velocity.

### Step 2: Identify the relevant concepts
Since the cart is moving with a constant velocity, the acceleration of the cart is zero. According to Newton's second law, if the acceleration is zero, the net force acting on the cart must also be zero. This means that the external force applied to the cart must balance the thrust force due to the sand being dropped into it.

### Step 3: Calculate the thrust force
The thrust force can be calculated using the formula:
\[ F_{\text{thrust}} = V_{\text{relative}} \times \frac{dm}{dt} \]
where:
- \( V_{\text{relative}} \) is the velocity of the cart (20 m/s).
- \( \frac{dm}{dt} \) is the rate at which mass is being added to the cart (50 kg/min).

### Step 4: Convert the mass flow rate to kg/s
Since the mass flow rate is given in kg/min, we need to convert it to kg/s:
\[ \frac{dm}{dt} = \frac{50 \text{ kg/min}}{60 \text{ s/min}} = \frac{50}{60} \text{ kg/s} = \frac{5}{6} \text{ kg/s} \]

### Step 5: Substitute the values into the thrust force equation
Now we can substitute the values into the thrust force equation:
\[ F_{\text{thrust}} = 20 \text{ m/s} \times \frac{5}{6} \text{ kg/s} \]
\[ F_{\text{thrust}} = \frac{100}{6} \text{ N} \]

### Step 6: Simplify the thrust force
Now, simplify \( \frac{100}{6} \):
\[ F_{\text{thrust}} = 16.67 \text{ N} \]

### Step 7: Conclusion
Since the net force must be zero for the cart to move with constant velocity, the external force required to move the cart is equal to the thrust force:
\[ F_{\text{external}} = F_{\text{thrust}} = 16.67 \text{ N} \]

Thus, the force required to move the cart with constant velocity is approximately **16.67 N**.

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