A body of mass 1 kg is acclerated uniformly from rest to a speed of 5 m/s in 4 sec. What is the instantaneous power delivered to the body at time t? (Assume `t lt 4` sec.)

Options

A. `25/16 t`
B. t
C. `16/25` t
D. `4/5` t

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the acceleration of the body
The body starts from rest and accelerates uniformly to a speed of 5 m/s in 4 seconds. We can use the formula for acceleration:

\[ a = \frac{v - u}{t} \]

Where:
- \( v = 5 \, \text{m/s} \) (final velocity)
- \( u = 0 \, \text{m/s} \) (initial velocity, since it starts from rest)
- \( t = 4 \, \text{s} \)

Substituting the values:

\[ a = \frac{5 \, \text{m/s} - 0 \, \text{m/s}}{4 \, \text{s}} = \frac{5}{4} \, \text{m/s}^2 \]

### Step 2: Calculate the force acting on the body
Using Newton's second law, the force \( F \) can be calculated as:

\[ F = m \cdot a \]

Where:
- \( m = 1 \, \text{kg} \) (mass of the body)
- \( a = \frac{5}{4} \, \text{m/s}^2 \) (acceleration from Step 1)

Substituting the values:

\[ F = 1 \, \text{kg} \cdot \frac{5}{4} \, \text{m/s}^2 = \frac{5}{4} \, \text{N} \]

### Step 3: Determine the velocity at time \( t \)
The velocity \( v \) at any time \( t \) can be calculated using the same formula for velocity:

\[ v = u + a \cdot t \]

Substituting the known values:

\[ v = 0 + \left(\frac{5}{4} \, \text{m/s}^2\right) \cdot t = \frac{5}{4} t \, \text{m/s} \]

### Step 4: Calculate the instantaneous power delivered to the body
The instantaneous power \( P \) can be calculated using the formula:

\[ P = F \cdot v \]

Substituting the values of force \( F \) and velocity \( v \):

\[ P = \left(\frac{5}{4} \, \text{N}\right) \cdot \left(\frac{5}{4} t \, \text{m/s}\right) \]

Calculating this gives:

\[ P = \frac{5}{4} \cdot \frac{5}{4} t = \frac{25}{16} t \, \text{W} \]

### Final Result
Thus, the instantaneous power delivered to the body at time \( t \) is:

\[ P(t) = \frac{25}{16} t \, \text{W} \]

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