A particle of mass M starting from rest undergoes uniform acceleration. If the speed acquired in time T is V, the power delivered to the particle is -

Options

A. `(1)/(2)(MV^(2))/(T^(2))`
B. `(MV^(2))/(T^(2))`
C. `(1)/(2)(MV^(2))/(T)`
D. `(MV^(2))/(T)`

Answer

To find the power delivered to a particle of mass \( M \) that starts from rest and undergoes uniform acceleration, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the initial conditions**:
The particle starts from rest, so the initial velocity \( u = 0 \).

2. **Use the equation of motion to find acceleration**:
The final velocity \( v \) after time \( T \) is given. Using the equation of motion:
\[ v = u + aT \]
Since \( u = 0 \), we have:
\[ v = aT \implies a = \frac{v}{T} \]

3. **Calculate the force acting on the particle**:
Using Newton's second law, the force \( F \) can be calculated as:
\[ F = M \cdot a = M \cdot \frac{v}{T} \]

4. **Determine the displacement of the particle**:
The displacement \( d \) during the time \( T \) can be calculated using the equation:
\[ d = uT + \frac{1}{2} a T^2 \]
Again, since \( u = 0 \):
\[ d = \frac{1}{2} a T^2 = \frac{1}{2} \left(\frac{v}{T}\right) T^2 = \frac{1}{2} vT \]

5. **Calculate the work done on the particle**:
Work done \( W \) is given by:
\[ W = F \cdot d \]
Substituting the values of \( F \) and \( d \):
\[ W = \left(M \cdot \frac{v}{T}\right) \cdot \left(\frac{1}{2} vT\right) = \frac{1}{2} M v^2 \]

6. **Determine the power delivered to the particle**:
Power \( P \) is defined as work done per unit time:
\[ P = \frac{W}{T} = \frac{\frac{1}{2} M v^2}{T} = \frac{1}{2} \frac{M v^2}{T} \]

### Final Answer:
The power delivered to the particle is:
\[ P = \frac{1}{2} \frac{M v^2}{T} \]