If the force acting on a body is inversely proportional to its speed, the kinetic energy of the body is

Options

A. constant
B. directly proportional to time
C. inversely proportional to time
D. directely proportional to square of time.

Explanation

To solve the problem, we need to analyze the relationship between force, speed, and kinetic energy given that the force acting on a body is inversely proportional to its speed. Let's break it down step by step.

### Step 1: Understand the relationship between force and speed
Given that the force \( F \) is inversely proportional to speed \( v \), we can express this relationship mathematically as:
\[ F = \frac{c}{v} \]
where \( c \) is a constant.

### Step 2: Relate force to acceleration
According to Newton's second law, force is also related to mass \( m \) and acceleration \( a \):
\[ F = m \cdot a \]
Since we have \( F = \frac{c}{v} \), we can equate the two expressions:
\[ m \cdot a = \frac{c}{v} \]

### Step 3: Express acceleration in terms of velocity
Acceleration \( a \) can be expressed as the derivative of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Substituting this into the equation gives:
\[ m \cdot \frac{dv}{dt} = \frac{c}{v} \]

### Step 4: Rearrange and separate variables
Rearranging the equation allows us to separate variables:
\[ m \cdot v \, dv = c \, dt \]

### Step 5: Integrate both sides
Now we integrate both sides. The left side integrates with respect to \( v \) and the right side with respect to \( t \):
\[ \int m \cdot v \, dv = \int c \, dt \]
This gives:
\[ \frac{m}{2} v^2 = ct + C \]
where \( C \) is the constant of integration.

### Step 6: Determine the constant of integration
Assuming the body starts from rest at \( t = 0 \) (i.e., \( v = 0 \)), we can find \( C \):
\[ \frac{m}{2} (0)^2 = c(0) + C \implies C = 0 \]
Thus, we have:
\[ \frac{m}{2} v^2 = ct \]

### Step 7: Relate to kinetic energy
The kinetic energy \( KE \) of the body is given by:
\[ KE = \frac{1}{2} m v^2 \]
From our earlier result, we can substitute:
\[ KE = ct \]

### Conclusion: Kinetic energy is proportional to time
This shows that the kinetic energy of the body is proportional to time:
\[ KE \propto t \]