A body of mass `3 kg` is under a constant force which causes a displacement `s` metre in it, given by the relation `s=(1)/(3)t^(2)`, where `t` is in seconds. Work done by the force in 2 seconds is

Options

A. `(19)/(5)J`
B. `(5)/(19)J`
C. `(3)/(8)J`
D. `(8)/(3)J`

Explanation

To solve the problem step by step, we need to find the work done by the force on a body of mass \(3 \, \text{kg}\) that is displaced according to the relation \(s = \frac{1}{3} t^2\) over a time interval of \(2\) seconds.

### Step 1: Determine the displacement at \(t = 2\) seconds.
Given the equation for displacement:
\[ s = \frac{1}{3} t^2 \]
Substituting \(t = 2\) seconds:
\[ s = \frac{1}{3} (2^2) = \frac{1}{3} \times 4 = \frac{4}{3} \, \text{m} \]

### Step 2: Calculate the velocity of the body.
The velocity \(v\) is the derivative of displacement \(s\) with respect to time \(t\):
\[ v = \frac{ds}{dt} = \frac{d}{dt} \left( \frac{1}{3} t^2 \right) = \frac{1}{3} \cdot 2t = \frac{2}{3} t \]
At \(t = 2\) seconds:
\[ v = \frac{2}{3} \times 2 = \frac{4}{3} \, \text{m/s} \]

### Step 3: Calculate the acceleration of the body.
The acceleration \(a\) is the derivative of velocity \(v\) with respect to time \(t\):
\[ a = \frac{dv}{dt} = \frac{d}{dt} \left( \frac{2}{3} t \right) = \frac{2}{3} \]
The acceleration is constant and does not depend on time.

### Step 4: Determine the force acting on the body.
Using Newton's second law, \(F = m \cdot a\):
\[ F = 3 \, \text{kg} \cdot \frac{2}{3} \, \text{m/s}^2 = 2 \, \text{N} \]

### Step 5: Calculate the work done by the force.
Work done \(W\) is given by the formula:
\[ W = F \cdot s \]
Substituting the values of force and displacement:
\[ W = 2 \, \text{N} \cdot \frac{4}{3} \, \text{m} = \frac{8}{3} \, \text{J} \]

### Final Answer:
The work done by the force in \(2\) seconds is \(\frac{8}{3} \, \text{J}\).

---