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 increase 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 \( K \) of a particle is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 2: Set Up the Equation
From the problem, we know:
\[ K = 4t^2 \]
Equating the two expressions for kinetic energy, we have:
\[ \frac{1}{2} mv^2 = 4t^2 \]

### Step 3: Solve for Velocity
Rearranging the equation to solve for \( v^2 \):
\[ mv^2 = 8t^2 \implies v^2 = \frac{8t^2}{m} \]
Taking the square root to find \( v \):
\[ v = \sqrt{\frac{8t^2}{m}} = \frac{2\sqrt{2}t}{\sqrt{m}} \]

### Step 4: Differentiate Velocity to Find Acceleration
To find the force, we first need to determine the acceleration \( a \). Acceleration is the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Differentiating \( v \):
\[ v = \frac{2\sqrt{2}}{\sqrt{m}} t \]
Thus,
\[ a = \frac{d}{dt}\left(\frac{2\sqrt{2}}{\sqrt{m}} t\right) = \frac{2\sqrt{2}}{\sqrt{m}} \]
This shows that the acceleration is constant.

### Step 5: Calculate the Force
Using Newton's second law, the force \( F \) acting on the particle is given by:
\[ F = ma \]
Substituting the expression for acceleration:
\[ F = m \cdot \frac{2\sqrt{2}}{\sqrt{m}} = 2\sqrt{2m} \]

### Conclusion
The force acting on the particle is:
\[ F = 2\sqrt{2m} \]
This force is constant since both \( m \) and \( \sqrt{2} \) are constants.

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