The kinetic energy of a body is given by equation `K=(2t^(3))` Joule (t in sec).The correct graph shown between applied force (F) and time (t) is ………….

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Answer

To solve the problem, we need to determine the relationship between the kinetic energy \( K \) of a body and the applied force \( F \) over time \( t \). The kinetic energy is given by the equation:

\[ K = 2t^3 \quad \text{(in Joules)} \]

### Step 1: Find the expression for velocity
The kinetic energy \( K \) is related to the velocity \( v \) of the body by the formula:

\[ K = \frac{1}{2} mv^2 \]

where \( m \) is the mass of the body. Rearranging this gives:

\[ v^2 = \frac{2K}{m} \]

Substituting the expression for \( K \):

\[ v^2 = \frac{2(2t^3)}{m} = \frac{4t^3}{m} \]

Taking the square root to find \( v \):

\[ v = \sqrt{\frac{4t^3}{m}} = \frac{2t^{3/2}}{\sqrt{m}} \]

### Step 2: Find the expression for acceleration
The acceleration \( a \) is the time derivative of velocity \( v \):

\[ a = \frac{dv}{dt} \]

Using the expression for \( v \):

\[ a = \frac{d}{dt}\left(\frac{2t^{3/2}}{\sqrt{m}}\right) \]

Using the power rule of differentiation:

\[ a = \frac{2}{\sqrt{m}} \cdot \frac{3}{2} t^{1/2} = \frac{3t^{1/2}}{\sqrt{m}} \]

### Step 3: Find the expression for force
Using Newton's second law, the force \( F \) is given by:

\[ F = ma \]

Substituting the expression for \( a \):

\[ F = m \cdot \frac{3t^{1/2}}{\sqrt{m}} = 3\sqrt{m} \cdot t^{1/2} \]

### Step 4: Analyze the relationship between force and time
From the equation \( F = 3\sqrt{m} \cdot t^{1/2} \), we see that force \( F \) is proportional to \( t^{1/2} \). This means that as time \( t \) increases, the force \( F \) increases with the square root of time.

### Step 5: Determine the graph
The graph of \( F \) versus \( t \) will be a curve that starts at the origin and rises gradually, resembling the shape of a square root function.

### Conclusion
Based on the analysis, the correct graph between applied force \( F \) and time \( t \) will show a curve that increases with the square root of time.