Kinetic energy of a particle moving in a straight line is proportional to the time t. The magnitude of the force acting on the particle is :

Options

A. Directly proportional to the speed of the particle
B. Directly proportional to `sqrt(t)`
C. Inversely proportional to the speed of the particle
D. Inversely proportional to `sqrt(t)`

Explanation

To solve the problem, we need to analyze the relationship between kinetic energy, velocity, and force. Here’s a step-by-step breakdown:

### Step 1: Understand the relationship between kinetic energy and time
Given that the kinetic energy (KE) of a particle is proportional to time \( t \), we can express this as:
\[ KE = ct \]
where \( c \) is a constant.

### Step 2: Relate kinetic energy to velocity
The kinetic energy of a particle is also given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 3: Set the two expressions for kinetic energy equal
From the two expressions for kinetic energy, we can set them equal to each other:
\[ \frac{1}{2} mv^2 = ct \]

### Step 4: Solve for velocity
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{2ct}{m} \]
Taking the square root gives us:
\[ v = \sqrt{\frac{2c}{m}} \sqrt{t} \]
Let \( C_0 = \sqrt{\frac{2c}{m}} \), then:
\[ v = C_0 \sqrt{t} \]

### Step 5: Differentiate velocity to find acceleration
To find the acceleration \( a \), we differentiate \( v \) with respect to time \( t \):
\[ a = \frac{dv}{dt} = C_0 \cdot \frac{1}{2} t^{-1/2} = \frac{C_0}{2\sqrt{t}} \]

### Step 6: Relate acceleration to force
The force \( F \) acting on the particle is given by Newton's second law:
\[ F = ma \]
Substituting the expression for acceleration:
\[ F = m \cdot \frac{C_0}{2\sqrt{t}} = \frac{mC_0}{2\sqrt{t}} \]

### Step 7: Conclude the relationship of force with time
From the expression for force, we see that:
\[ F \propto \frac{1}{\sqrt{t}} \]
This shows that the magnitude of the force acting on the particle is inversely proportional to the square root of time.

### Final Answer
The magnitude of the force acting on the particle is inversely proportional to \( \sqrt{t} \).

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