If the kinetic energy of a body is directly proportional to time t, the magnitude of the force acting on the body is

Options

A. directly proportional to `sqrt(t)`
B. inversely proportional to `sqrt(t)`
C. directly proportional to the speed of the body
D. inversely proportional to the speed of the body

Answer

To solve the problem step by step, we will analyze the relationship between kinetic energy, velocity, acceleration, and force.

### Step 1: Establish the relationship between kinetic energy and time
Given that the kinetic energy \( K \) of a body is directly proportional to time \( t \), we can express this mathematically as:
\[ K = C \cdot t \]
where \( C \) is a constant.

### Step 2: Relate kinetic energy to velocity
The kinetic energy of a body is also given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the body and \( v \) is its velocity. Setting the two expressions for kinetic energy equal to each other, we have:
\[ \frac{1}{2} mv^2 = C \cdot t \]

### Step 3: Solve for velocity
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{2C \cdot t}{m} \]
Taking the square root of both sides gives us the velocity \( v \):
\[ v = \sqrt{\frac{2C \cdot t}{m}} \]

### Step 4: Find acceleration
Acceleration \( a \) is defined as the rate of change of velocity with respect to time. Therefore, we differentiate \( v \) with respect to \( t \):
\[ a = \frac{dv}{dt} \]
Using the expression for \( v \):
\[ v = \sqrt{\frac{2C}{m}} \cdot t^{1/2} \]
Differentiating this with respect to \( t \):
\[ a = \sqrt{\frac{2C}{m}} \cdot \frac{1}{2} t^{-1/2} \]
This simplifies to:
\[ a = \frac{\sqrt{2C}}{2\sqrt{m}} \cdot \frac{1}{\sqrt{t}} \]

### Step 5: Relate force to mass and acceleration
According to Newton's second law, the force \( F \) acting on the body is given by:
\[ F = m \cdot a \]
Substituting the expression for acceleration:
\[ F = m \cdot \frac{\sqrt{2C}}{2\sqrt{m}} \cdot \frac{1}{\sqrt{t}} \]
This simplifies to:
\[ F = \frac{\sqrt{2Cm}}{2\sqrt{t}} \]

### Step 6: Determine the relationship between force and time
From the expression for force, we can see that:
\[ F \propto \frac{1}{\sqrt{t}} \]
This indicates that the force is inversely proportional to the square root of time.

### Conclusion
Thus, the magnitude of the force acting on the body is inversely proportional to the square root of time.