An object of mass m, initially at rest under the action of a constant force F attains a velocity v in time t. Then, the average power supplied to mass is

Options

A. `(mv^(2))/(2t)`
B. `(Fv)/(2)`
C. Both are correct
D. Both are wrong

Explanation

To solve the problem of finding the average power supplied to an object of mass \( m \) that is initially at rest and attains a velocity \( v \) under the action of a constant force \( F \) in time \( t \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Average Power**:
Average power \( P_{\text{avg}} \) is defined as the work done \( W \) divided by the time \( t \):
\[ P_{\text{avg}} = \frac{W}{t} \]

2. **Determine the Work Done**:
The work done on the object is equal to the change in kinetic energy. Since the object starts from rest, the initial kinetic energy is 0. The final kinetic energy when the object reaches velocity \( v \) is given by:
\[ KE = \frac{1}{2} m v^2 \]
Therefore, the work done \( W \) is:
\[ W = \frac{1}{2} m v^2 \]

3. **Substitute Work Done into the Average Power Formula**:
Now, substituting the expression for work done into the average power formula:
\[ P_{\text{avg}} = \frac{W}{t} = \frac{\frac{1}{2} m v^2}{t} \]
This simplifies to:
\[ P_{\text{avg}} = \frac{m v^2}{2t} \]

4. **Relate Force to Acceleration**:
From Newton's second law, we know that the force \( F \) acting on the object is related to its mass and acceleration \( a \) by:
\[ F = ma \]
Since the object accelerates from rest to velocity \( v \) in time \( t \), we can express acceleration as:
\[ a = \frac{v}{t} \]

5. **Substitute Acceleration into the Force Equation**:
Substituting \( a \) into the force equation gives:
\[ F = m \left(\frac{v}{t}\right) \]

6. **Express Average Power in Terms of Force**:
Now, we can express the average power in terms of the force:
\[ P_{\text{avg}} = \frac{m v^2}{2t} = \frac{F v}{2} \]

### Final Result:
Thus, the average power supplied to the mass is:
\[ P_{\text{avg}} = \frac{m v^2}{2t} \quad \text{or} \quad P_{\text{avg}} = \frac{F v}{2} \]