Kinetic energy of a particle moving in a straight line varies with time `t` as `K = 4t^(2)`. The force acting on the particle

Options

A. is constant
B. is increasing
C. is decreasing
D. first increases and then decreases

Explanation

To find the force acting on a particle whose kinetic energy varies with time as \( K = 4t^2 \), we can follow these steps:

### Step 1: Relate Kinetic Energy to Velocity
The kinetic energy \( K \) is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
Substituting the given expression for kinetic energy:
\[ \frac{1}{2} mv^2 = 4t^2 \]

### Step 2: Solve for Velocity
Rearranging the equation to solve for \( v^2 \):
\[ mv^2 = 8t^2 \]
\[ v^2 = \frac{8t^2}{m} \]
Taking the square root of both sides gives:
\[ v = \sqrt{\frac{8t^2}{m}} = \frac{2\sqrt{2}t}{\sqrt{m}} \]

### Step 3: Determine Acceleration
Since the velocity \( v \) is a function of time \( t \), we can find the acceleration \( a \) by differentiating the velocity with respect to time:
\[ a = \frac{dv}{dt} = \frac{d}{dt}\left(\frac{2\sqrt{2}t}{\sqrt{m}}\right) = \frac{2\sqrt{2}}{\sqrt{m}} \]

### Step 4: Calculate Force
Using Newton's second law, the force \( F \) acting on the particle is given by:
\[ F = ma \]
Substituting the expression for acceleration:
\[ F = m \left(\frac{2\sqrt{2}}{\sqrt{m}}\right) \]
Simplifying this gives:
\[ F = 2\sqrt{2} \sqrt{m} \]

### Final Result
Thus, the force acting on the particle is:
\[ F = 2\sqrt{2} \sqrt{m} \]

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