A body of mass m accelerates uniformly from rest to `v_1` in time `t_1`. As a function of time t, the instantaneous power delivered to the body is

Options

A. `(mv_1t)/t_1`
B. `(mv_1^2t)/t_1`
C. `(mv_1t^2)/t_1`
D. `(mv_1^2 t)/(t_1^2)`

Answer

To solve the problem, we need to find the instantaneous power delivered to a body of mass \( m \) that accelerates uniformly from rest to a velocity \( v_1 \) in a time \( t_1 \).

### Step-by-step Solution:

1. **Understanding Power**:
Power (\( P \)) is defined as the rate at which work is done or energy is transferred. Mathematically, it can be expressed as:
\[ P = F \cdot v \]
where \( F \) is the force applied and \( v \) is the velocity of the body.

2. **Finding Acceleration**:
Since the body starts from rest and accelerates uniformly to \( v_1 \) in time \( t_1 \), we can use the first equation of motion:
\[ v = u + at \]
Here, the initial velocity \( u = 0 \), so:
\[ v_1 = 0 + a t_1 \implies a = \frac{v_1}{t_1} \]
This gives us the acceleration \( a \) as a function of \( v_1 \) and \( t_1 \).

3. **Finding Velocity as a Function of Time**:
We can express the velocity \( v \) at any time \( t \) during the acceleration:
\[ v = u + at = 0 + \left(\frac{v_1}{t_1}\right) t = \frac{v_1}{t_1} t \]
This is our equation for velocity as a function of time \( t \).

4. **Finding Force**:
The force \( F \) acting on the body can be calculated using Newton's second law:
\[ F = ma \]
Substituting the expression for acceleration:
\[ F = m \left(\frac{v_1}{t_1}\right) \]

5. **Calculating Instantaneous Power**:
Now we can substitute the expressions for force and velocity into the power equation:
\[ P = F \cdot v = \left(m \frac{v_1}{t_1}\right) \cdot \left(\frac{v_1}{t_1} t\right) \]
Simplifying this gives:
\[ P = m \frac{v_1^2}{t_1^2} t \]

6. **Final Expression for Power**:
Thus, the instantaneous power delivered to the body as a function of time \( t \) is:
\[ P(t) = \frac{m v_1^2}{t_1^2} t \]

### Conclusion:
The instantaneous power delivered to the body is given by:
\[ P(t) = \frac{m v_1^2}{t_1^2} t \]