A body of mass `m`, accelerates uniformly from rest to `V_(1)` in time `t_(1)`. The instantaneous power delivered to the body as a function of time `t` is.

Options

A. `(mv_(1)t^(2))/(t_(1))`
B. `(mv_(1)^(2)t)/(t_(1)^(2))`
C. `(mv_(1)t)/(t_(1))`
D. `(mv_(1)^(2) t)/(t_(1))`

Explanation

To find the instantaneous power delivered to a body of mass `m` that accelerates uniformly from rest to a velocity `V_(1)` in time `t_(1)`, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the acceleration (a)**:
Since the body starts from rest (initial velocity `u = 0`) and reaches a final velocity `V_(1)` in time `t_(1)`, we can calculate the acceleration using the formula:
\[ a = \frac{V_{1} - u}{t_{1}} = \frac{V_{1} - 0}{t_{1}} = \frac{V_{1}}{t_{1}} \]

2. **Calculate the instantaneous velocity (v) at time t**:
Using the first equation of motion, we can express the velocity at any time `t` as:
\[ v = u + at = 0 + a \cdot t = a \cdot t \]
Substituting the value of `a` from step 1:
\[ v = \left(\frac{V_{1}}{t_{1}}\right) \cdot t \]

3. **Determine the force (F)**:
The force acting on the body can be calculated using Newton's second law:
\[ F = m \cdot a \]
Substituting the value of `a`:
\[ F = m \cdot \left(\frac{V_{1}}{t_{1}}\right) \]

4. **Calculate the instantaneous power (P)**:
The instantaneous power delivered to the body is given by the product of force and velocity:
\[ P = F \cdot v \]
Substituting the expressions for `F` and `v`:
\[ P = \left(m \cdot \frac{V_{1}}{t_{1}}\right) \cdot \left(\frac{V_{1}}{t_{1}} \cdot t\right) \]
Simplifying this expression:
\[ P = m \cdot \frac{V_{1}^2}{t_{1}^2} \cdot t \]

### Final Expression:
Thus, the instantaneous power delivered to the body as a function of time `t` is:
\[ P(t) = m \cdot \frac{V_{1}^2}{t_{1}^2} \cdot t \]