Work done in time t on a body of mass m which is accelerated from rest to a speed v in time `t_(1)` as a function of time t is given by

Options

A. `(1)/(2)m(v^(2))/(t_(1)^(2))t^(2)`
B. `(1)/(2)((mv)/(t_(1)))^(2)t^(2)`
C. `m(v)/(t_(1))t^(2)`
D. `(1)/(2)m(v)/(t)t^(2)`

Explanation

To solve the problem of finding the work done on a body of mass \( m \) that is accelerated from rest to a speed \( v \) in time \( t_1 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- The body has a mass \( m \).
- It starts from rest, so the initial velocity \( u = 0 \).
- It reaches a final speed \( v \) in time \( t_1 \).

2. **Calculate Acceleration**:
- Using the equation of motion:
\[ v = u + at \]
Since \( u = 0 \), we have:
\[ v = at_1 \]
- Rearranging gives us the acceleration \( a \):
\[ a = \frac{v}{t_1} \]

3. **Calculate Displacement**:
- The displacement \( s \) during the time \( t_1 \) can be calculated using the formula:
\[ s = ut + \frac{1}{2} a t^2 \]
Again, since \( u = 0 \):
\[ s = \frac{1}{2} a t_1^2 \]
- Substituting the value of \( a \):
\[ s = \frac{1}{2} \left(\frac{v}{t_1}\right) t_1^2 = \frac{1}{2} vt_1 \]

4. **Calculate Work Done**:
- The work done \( W \) on the body is given by:
\[ W = \text{Force} \times \text{Displacement} \]
- The force \( F \) can be calculated using Newton's second law:
\[ F = ma = m \left(\frac{v}{t_1}\right) \]
- Therefore, substituting for force and displacement:
\[ W = F \cdot s = \left(m \frac{v}{t_1}\right) \cdot \left(\frac{1}{2} vt_1\right) \]
- Simplifying this gives:
\[ W = \frac{1}{2} mv \cdot v = \frac{1}{2} mv^2 \]

5. **Final Expression**:
- The work done in time \( t \) on the body is:
\[ W = \frac{1}{2} mv^2 \]