The intial velocity of a particle moving along x axis is u (at t = 0 and x = 0) and its acceleration a is given by a = kx. Which of the following equation is correct between its velocity (v) and position (x)?

Options

A. `v^(2)=y^(2)=2kx`
B. `v^(2)=u^(2)+2kx^(2)`
C. `v^(2)=u^(2)+kx^(2)`
D. `v^(2)+u^(2)=2kx`

Answer

To solve the problem, we need to derive the relationship between the velocity \( v \) and the position \( x \) of a particle given that its acceleration \( a \) is proportional to its position \( x \), specifically \( a = kx \).

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Initial velocity \( u \) at \( t = 0 \) and \( x = 0 \).
- Acceleration \( a = kx \).

2. **Relate Acceleration to Velocity and Position:**
- We know that acceleration can be expressed in two ways:
\[ a = \frac{dv}{dt} \quad \text{(1)} \]
and
\[ a = v \frac{dv}{dx} \quad \text{(2)} \]
- Since both expressions represent acceleration, we can set them equal to each other:
\[ \frac{dv}{dt} = v \frac{dv}{dx} \]

3. **Substituting the Expression for Acceleration:**
- From the problem, we have \( a = kx \). Therefore, we can substitute \( a \) in equation (1):
\[ kx = v \frac{dv}{dx} \]

4. **Rearranging the Equation:**
- Rearranging gives:
\[ v \, dv = kx \, dx \]

5. **Integrating Both Sides:**
- Now we integrate both sides. The left side integrates from \( u \) to \( v \) and the right side from \( 0 \) to \( x \):
\[ \int_{u}^{v} v \, dv = \int_{0}^{x} kx \, dx \]

6. **Calculating the Integrals:**
- The left-hand side:
\[ \frac{v^2}{2} \bigg|_{u}^{v} = \frac{v^2}{2} - \frac{u^2}{2} \]
- The right-hand side:
\[ k \int_{0}^{x} x \, dx = k \left( \frac{x^2}{2} \right) \bigg|_{0}^{x} = k \frac{x^2}{2} \]

7. **Setting the Two Integrals Equal:**
- Now we equate both results:
\[ \frac{v^2}{2} - \frac{u^2}{2} = k \frac{x^2}{2} \]

8. **Simplifying the Equation:**
- Multiply through by 2 to eliminate the fractions:
\[ v^2 - u^2 = kx^2 \]
- Rearranging gives:
\[ v^2 = u^2 + kx^2 \]

### Final Result:
The correct relationship between the velocity \( v \) and the position \( x \) is:
\[ v^2 = u^2 + kx^2 \]