If the kinetic energy of body is directly proportional to time `t`, the magnitude of force acting on the body is (i) directly proportional to `sqrtt`
(ii) inversely proportional to `sqrtt`
(iii) directly proportional to the speed of the body.
(iv) inversely proportional to the speed of the body.

Options

A. (i),(ii)
B. (i),(iii)
C. (ii),(iv)
D. (i),(iv)

Explanation

To solve the problem, we need to analyze the relationship between kinetic energy, time, force, and speed. Let's break it down step by step.

### Step 1: Understanding 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 relationship mathematically as:
\[ K = \mu t \]
where \( \mu \) is a constant of proportionality.

### Step 2: Relating 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 gives us:
\[ \frac{1}{2} mv^2 = \mu t \]

### Step 3: Solving for velocity
From the equation above, we can solve for \( v^2 \):
\[ v^2 = \frac{2\mu t}{m} \]
Taking the square root to find \( v \):
\[ v = \sqrt{\frac{2\mu}{m} t} \]
This shows that velocity \( v \) is directly proportional to \( \sqrt{t} \).

### Step 4: Finding acceleration
Acceleration \( a \) is the rate of change of velocity with respect to time, which can be expressed as:
\[ a = \frac{dv}{dt} \]
Differentiating \( v = \sqrt{\frac{2\mu}{m} t} \):
\[ a = \frac{d}{dt}\left(\sqrt{\frac{2\mu}{m}} \cdot t^{1/2}\right) = \sqrt{\frac{2\mu}{m}} \cdot \frac{1}{2} t^{-1/2} \]
Thus, we have:
\[ a = \frac{\sqrt{2\mu}}{2\sqrt{m}} \cdot \frac{1}{\sqrt{t}} \]

### Step 5: Finding force
Using Newton's second law, the force \( F \) acting on the body is given by:
\[ F = ma \]
Substituting the expression for acceleration:
\[ F = m \cdot \left(\frac{\sqrt{2\mu}}{2\sqrt{m}} \cdot \frac{1}{\sqrt{t}}\right) \]
This simplifies to:
\[ F = \frac{\sqrt{2\mu} \cdot m^{1/2}}{2\sqrt{t}} \]
Thus, we can conclude that:
\[ F \propto \frac{1}{\sqrt{t}} \]
This indicates that the force is inversely proportional to \( \sqrt{t} \).

### Step 6: Relating force to speed
From our earlier steps, we found that:
\[ v \propto \sqrt{t} \]
Therefore, we can express \( \sqrt{t} \) in terms of \( v \):
\[ \sqrt{t} \propto v \]
Thus, substituting this into our expression for force:
\[ F \propto \frac{1}{\sqrt{t}} \propto \frac{1}{v} \]
This means that the force is inversely proportional to the speed of the body.

### Conclusion
From our analysis, we can conclude that:
- The magnitude of force acting on the body is inversely proportional to the speed of the body.

### Final Answer
The correct option is (iv) inversely proportional to the speed of the body.