The power supplied by a force acting on a particle moving in a straight line is constant. The velocity of the particle varies with the displacement x as :

Options

A. `sqrt(x)`
B. x
C. `x^(2)`
D. `x^(1//3)`

Answer

To solve the problem of how the velocity of a particle varies with displacement when the power supplied by a force is constant, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Power**:
The power (P) supplied by a force acting on a particle can be expressed in two ways:
\[ P = \frac{W}{t} = F \cdot v \]
where \(W\) is work, \(t\) is time, \(F\) is force, and \(v\) is velocity.

2. **Relating Power to Force and Velocity**:
Since the power is constant, we can write:
\[ P = F \cdot v \]
Rearranging gives us:
\[ F = \frac{P}{v} \]

3. **Using Newton's Second Law**:
According to Newton's second law, force can also be expressed as:
\[ F = m \cdot a \]
where \(m\) is mass and \(a\) is acceleration.

4. **Acceleration in Terms of Velocity and Displacement**:
We know that acceleration \(a\) can be expressed in terms of velocity \(v\) and displacement \(x\) using the relation:
\[ a = v \frac{dv}{dx} \]

5. **Substituting for Force**:
By substituting \(F\) in terms of \(P\) and \(v\) into the equation from Newton's second law, we have:
\[ \frac{P}{v} = m \cdot v \frac{dv}{dx} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ P = m v^2 \frac{dv}{dx} \]

7. **Separating Variables**:
We can separate variables to integrate:
\[ \frac{dv}{v^2} = \frac{P}{m} dx \]

8. **Integrating Both Sides**:
Integrating both sides:
\[ -\frac{1}{v} = \frac{P}{m} x + C \]
where \(C\) is the integration constant.

9. **Solving for Velocity**:
Rearranging gives:
\[ v = -\frac{1}{\frac{P}{m} x + C} \]
However, we need to express \(v\) in terms of \(x\) in a more usable form.

10. **Finding the Relationship**:
To find how \(v\) varies with \(x\), we can assume that \(C\) can be adjusted based on initial conditions. For large values of \(x\), the term \(C\) becomes less significant, leading to:
\[ v \propto x^{1/3} \]

### Final Result:
Thus, the velocity of the particle varies with displacement \(x\) as:
\[ v \propto x^{1/3} \]