The potential energy of a particle of mass m is given by `U=(1)/(2)kx^(2)` for `x lt 0` and U = 0 for `x ge 0`. If total mechanical energy of the particle is E. Then its speed at `x = sqrt((2E)/(k))` is

Options

A. zero
B. `sqrt((2E)/(m))`
C. `sqrt((E)/(m))`
D. `sqrt((E)/(2m))`

Answer

To solve the problem, we need to find the speed of a particle at a specific position given its potential energy function and total mechanical energy.

### Step-by-Step Solution:

1. **Understand the Potential Energy Function**:
The potential energy \( U \) is given by:
\[ U = \frac{1}{2} k x^2 \quad \text{for } x < 0 \]
and
\[ U = 0 \quad \text{for } x \geq 0 \]

2. **Total Mechanical Energy**:
The total mechanical energy \( E \) of the particle is the sum of its kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]

3. **Determine Potential Energy at the Given Position**:
We need to find the potential energy when \( x = \sqrt{\frac{2E}{k}} \). Since \( x \) is positive, we use the second part of the potential energy function:
\[ U = 0 \quad \text{for } x \geq 0 \]

4. **Calculate Kinetic Energy**:
Substitute \( U \) into the total mechanical energy equation:
\[ E = K + 0 \implies K = E \]

5. **Relate Kinetic Energy to Speed**:
The kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} m v^2 \]
Setting this equal to \( E \):
\[ E = \frac{1}{2} m v^2 \]

6. **Solve for Speed \( v \)**:
Rearranging the equation to find \( v \):
\[ v^2 = \frac{2E}{m} \]
Taking the square root gives:
\[ v = \sqrt{\frac{2E}{m}} \]

7. **Conclusion**:
Therefore, the speed of the particle at \( x = \sqrt{\frac{2E}{k}} \) is:
\[ v = \sqrt{\frac{2E}{m}} \]