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))`

Explanation

To solve the problem, we need to find the speed of a particle at a specific position given its potential energy and total mechanical energy. Let's break it down step by step.

### Step 1: Understand the potential energy function
The potential energy \( U \) of the particle is defined as:
- \( U = \frac{1}{2} k x^2 \) for \( x < 0 \)
- \( U = 0 \) for \( x \geq 0 \)

### Step 2: Identify the total mechanical energy
The total mechanical energy \( E \) of the particle is the sum of its potential energy \( U \) and kinetic energy \( K \):
\[ E = U + K \]

### Step 3: Analyze the position given in the problem
We need to find the speed of the particle at \( x = \sqrt{\frac{2E}{k}} \). Since this value of \( x \) is positive, we can use the condition for potential energy:
- At \( x = \sqrt{\frac{2E}{k}} \), since \( x \geq 0 \), the potential energy \( U = 0 \).

### Step 4: Substitute into the total mechanical energy equation
Substituting \( U = 0 \) into the total mechanical energy equation gives:
\[ E = 0 + K \implies K = E \]

### Step 5: Express kinetic energy in terms of speed
The kinetic energy \( K \) can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
Setting this equal to the total mechanical energy \( E \):
\[ E = \frac{1}{2} mv^2 \]

### Step 6: Solve for speed \( v \)
Rearranging the equation to solve for \( v \):
\[ \frac{1}{2} mv^2 = E \implies mv^2 = 2E \implies v^2 = \frac{2E}{m} \implies v = \sqrt{\frac{2E}{m}} \]

### Final Answer
Thus, the speed of the particle at \( x = \sqrt{\frac{2E}{k}} \) is:
\[ v = \sqrt{\frac{2E}{m}} \]