A particle moves in a straight line with acceleration proportional to `x^2`, ( Where x is displacement). The gain of kinetic energy for any displacement is proportional to

Options

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

Answer

To solve the problem, we need to analyze the motion of a particle under the influence of an acceleration that is proportional to the square of its displacement. We will derive the relationship between the gain in kinetic energy and the displacement.

### Step-by-Step Solution:

1. **Understanding the Acceleration**:
We know that the acceleration \( a \) is proportional to the square of the displacement \( x \). We can express this as:
\[ a = kx^2 \]
where \( k \) is a proportionality constant.

2. **Relating Acceleration to Velocity**:
We can relate acceleration to velocity using the chain rule. Recall that acceleration \( a \) can be expressed as:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \frac{dv}{dx} \]
Therefore, we can write:
\[ v \frac{dv}{dx} = kx^2 \]

3. **Separating Variables**:
Rearranging the equation gives:
\[ v \, dv = kx^2 \, dx \]

4. **Integrating Both Sides**:
We integrate both sides to find the relationship between velocity and displacement. The left side integrates to:
\[ \int v \, dv = \frac{v^2}{2} + C \]
The right side integrates to:
\[ \int kx^2 \, dx = \frac{kx^3}{3} + C' \]
Thus, we have:
\[ \frac{v^2}{2} = \frac{kx^3}{3} + C \]

5. **Finding the Change in Kinetic Energy**:
The change in kinetic energy \( \Delta KE \) as the particle moves from an initial position \( x_0 \) to a final position \( x \) can be expressed as:
\[ \Delta KE = \frac{v^2}{2} - \frac{u^2}{2} \]
where \( u \) is the initial velocity at position \( x_0 \). From our previous integration, we can express this as:
\[ \Delta KE = \left(\frac{kx^3}{3} - \frac{kx_0^3}{3}\right) \]

6. **Proportionality**:
Thus, we can conclude that:
\[ \Delta KE \propto (x^3 - x_0^3) \]
This shows that the gain in kinetic energy is proportional to the change in the cube of the displacement.

### Final Answer:
The gain in kinetic energy for any displacement is proportional to \( \Delta x^3 \).