A body initially at rest is acted upon by a constant force. The rate of change of its kinetic energy varies

Options

A. linearly with square root of time
B. linearly with time
C. linearly with square of time
D. inversely with time

Explanation

To solve the problem step by step, we will analyze the situation where a body initially at rest is acted upon by a constant force. We need to determine how the rate of change of its kinetic energy varies.

### Step 1: Understand the initial conditions
The body is initially at rest, which means its initial velocity \( u = 0 \).

### Step 2: Apply Newton's second law
According to Newton's second law, the force \( F \) acting on the body is related to its mass \( m \) and acceleration \( a \) by the equation:
\[ F = ma \]

### Step 3: Relate acceleration to velocity
Since the body starts from rest and is acted upon by a constant force, its velocity \( v \) at any time \( t \) can be expressed as:
\[ v = u + at = 0 + at = at \]

### Step 4: Determine the expression for kinetic energy
The kinetic energy \( KE \) of the body is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the expression for \( v \):
\[ KE = \frac{1}{2} m(at)^2 = \frac{1}{2} ma^2 t^2 \]

### Step 5: Calculate the rate of change of kinetic energy
To find the rate of change of kinetic energy with respect to time, we differentiate \( KE \) with respect to \( t \):
\[ \frac{d(KE)}{dt} = \frac{d}{dt}\left(\frac{1}{2} ma^2 t^2\right) = \frac{1}{2} ma^2 \cdot 2t = ma^2 t \]

### Step 6: Analyze the result
The expression \( ma^2 t \) indicates that the rate of change of kinetic energy is directly proportional to time \( t \).

### Conclusion
Thus, the rate of change of kinetic energy varies linearly with time. Therefore, the correct answer is:
**Option B: Linearly with time.**

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