A body of mass `m` is acceleratad uniformaly from rest to a speed `v` in a time `T` . The instanseous power delivered to the body as a function of time is given by

Options

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

Explanation

To solve the problem of finding the instantaneous power delivered to a body of mass \( m \) that is accelerated uniformly from rest to a speed \( v \) in a time \( T \), we can follow these steps:

### Step 1: Determine the acceleration
Since the body starts from rest, the initial velocity \( u = 0 \). The final velocity \( v \) is achieved in time \( T \). Using the equation of motion:

\[ v = u + aT \]

Substituting \( u = 0 \):

\[ v = aT \implies a = \frac{v}{T} \]

### Step 2: Calculate the instantaneous power
The instantaneous power \( P \) delivered to the body can be expressed as:

\[ P = F \cdot v \]

where \( F \) is the force applied and \( v \) is the velocity of the body at time \( t \).

### Step 3: Find the force using Newton's second law
According to Newton's second law, the force \( F \) is given by:

\[ F = ma \]

Substituting the expression for acceleration \( a \):

\[ F = m \cdot \frac{v}{T} \]

### Step 4: Express the velocity as a function of time
Since the body is accelerating uniformly, the velocity \( v(t) \) at any time \( t \) can be expressed as:

\[ v(t) = a t = \frac{v}{T} t \]

### Step 5: Substitute the force and velocity into the power equation
Now substitute \( F \) and \( v(t) \) into the power equation:

\[ P(t) = F \cdot v(t) = \left(m \cdot \frac{v}{T}\right) \cdot \left(\frac{v}{T} t\right) \]

### Step 6: Simplify the expression
Now simplify the expression for power:

\[ P(t) = m \cdot \frac{v}{T} \cdot \frac{v}{T} t = m \cdot \frac{v^2}{T^2} t \]

### Step 7: Final expression for instantaneous power
Thus, the instantaneous power delivered to the body as a function of time is:

\[ P(t) = \frac{m v^2}{T^2} t \]

### Conclusion
The expression for instantaneous power \( P(t) \) indicates that it is directly proportional to time \( t \), with the proportionality constant being \( \frac{m v^2}{T^2} \).

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