At time `t=0s` particle starts moving along the `x-` axis. If its kinetic energy increases uniformly with time `'t'`, the net force acting on it must be proportional to

Options

A. `sqrt(t0`
B. constant
C. `t`
D. `(1)/(sqrt(t))`

Explanation

To solve the problem step by step, we need to analyze the relationship between kinetic energy, momentum, and force acting on the particle.

### Step 1: Understand the relationship between kinetic energy and time
The problem states that the kinetic energy (KE) of the particle increases uniformly with time \( t \). We can express this as:
\[ KE = k \cdot t \]
where \( k \) is a constant.

### Step 2: Relate kinetic energy to momentum
The kinetic energy can also be expressed in terms of momentum \( p \) as:
\[ KE = \frac{p^2}{2m} \]
where \( m \) is the mass of the particle.

### Step 3: Equate the two expressions for kinetic energy
From the two equations for kinetic energy, we have:
\[ k \cdot t = \frac{p^2}{2m} \]
Rearranging gives:
\[ p^2 = 2m \cdot k \cdot t \]

### Step 4: Find the expression for momentum
Taking the square root of both sides, we find:
\[ p = \sqrt{2m \cdot k \cdot t} \]

### Step 5: Determine the force acting on the particle
According to Newton's second law, the net force \( F \) acting on the particle is the rate of change of momentum:
\[ F = \frac{dp}{dt} \]
Substituting the expression for momentum:
\[ F = \frac{d}{dt} \left( \sqrt{2m \cdot k \cdot t} \right) \]

### Step 6: Differentiate the momentum with respect to time
Using the chain rule, we differentiate:
\[ F = \frac{d}{dt} \left( \sqrt{2m \cdot k} \cdot t^{1/2} \right) = \sqrt{2m \cdot k} \cdot \frac{1}{2} t^{-1/2} \]
This simplifies to:
\[ F = \frac{\sqrt{2m \cdot k}}{2\sqrt{t}} \]

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

### Final Answer
The net force acting on the particle must be proportional to:
\[ F \propto \frac{1}{\sqrt{t}} \]