A body starts from rest and acquires velocity `V` in time `T`. The instantaneous power delivered to the body in time 't' proportional to

Options

A. `(V)/(T) t`
B. `(V^(2))/(T) t^(2)`
C. `(V^(2))/(T^(2)) t`
D. `(V^(2))/(T^(2)) t^(2)`

Answer

To solve the problem step by step, we will analyze the relationship between the instantaneous power delivered to a body, its velocity, and the time taken to acquire that velocity.

### Step 1: Understand the given information
- A body starts from rest and acquires a velocity \( V \) in time \( T \).
- We need to find the instantaneous power delivered to the body at a time \( t \).

### Step 2: Determine the acceleration
Since the body starts from rest and acquires velocity \( V \) in time \( T \), we can assume that the acceleration \( a \) is constant. The formula for acceleration is given by:
\[ a = \frac{V}{T} \]

### Step 3: Find the instantaneous velocity at time \( t \)
The instantaneous velocity \( v \) at any time \( t \) can be expressed as:
\[ v = a \cdot t = \left(\frac{V}{T}\right) \cdot t = \frac{V}{T} \cdot t \]

### Step 4: Calculate the instantaneous power
The instantaneous power \( P \) delivered to the body can be calculated using the formula:
\[ P = F \cdot v \]
where \( F \) is the force acting on the body. Since \( F = m \cdot a \) (Newton's second law), we can substitute \( a \) into the equation:
\[ F = m \cdot \frac{V}{T} \]
Thus, the power becomes:
\[ P = \left(m \cdot \frac{V}{T}\right) \cdot v \]
Substituting \( v \) from step 3:
\[ P = \left(m \cdot \frac{V}{T}\right) \cdot \left(\frac{V}{T} \cdot t\right) \]
This simplifies to:
\[ P = m \cdot \frac{V^2}{T^2} \cdot t \]

### Step 5: Determine the proportionality
From the expression derived for power:
\[ P \propto t \]
This indicates that the instantaneous power delivered to the body at time \( t \) is directly proportional to \( t \).

### Conclusion
The instantaneous power delivered to the body in time \( t \) is proportional to \( t \).