A block of mass m moving at a speed v compresses as spring thrugh a distance x before its speed is halved. Find the spring constant of the spring.

Explanation

To find the spring constant \( k \) of the spring when a block of mass \( m \) compresses the spring through a distance \( x \) before its speed is halved, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the initial and final states
- Initially, the block is moving with speed \( v \) and has kinetic energy given by:
\[ KE_{initial} = \frac{1}{2} mv^2 \]
- When the block compresses the spring by a distance \( x \), its speed becomes \( \frac{v}{2} \). The kinetic energy at this point is:
\[ KE_{final} = \frac{1}{2} m \left(\frac{v}{2}\right)^2 = \frac{1}{2} m \cdot \frac{v^2}{4} = \frac{1}{8} mv^2 \]

### Step 2: Write the energy conservation equation
- According to the conservation of energy, the initial kinetic energy minus the final kinetic energy is equal to the potential energy stored in the spring:
\[ KE_{initial} - KE_{final} = PE_{spring} \]
- The potential energy stored in the spring when compressed by distance \( x \) is given by:
\[ PE_{spring} = \frac{1}{2} k x^2 \]

### Step 3: Set up the equation
- Substitute the expressions for kinetic energy into the energy conservation equation:
\[ \frac{1}{2} mv^2 - \frac{1}{8} mv^2 = \frac{1}{2} k x^2 \]

### Step 4: Simplify the equation
- Combine the kinetic energy terms on the left:
\[ \frac{1}{2} mv^2 - \frac{1}{8} mv^2 = \frac{4}{8} mv^2 - \frac{1}{8} mv^2 = \frac{3}{8} mv^2 \]
- Now the equation becomes:
\[ \frac{3}{8} mv^2 = \frac{1}{2} k x^2 \]

### Step 5: Solve for the spring constant \( k \)
- Rearranging the equation to isolate \( k \):
\[ k = \frac{3 mv^2}{4 x^2} \]

### Final Answer
The spring constant \( k \) is given by:
\[ k = \frac{3mv^2}{4x^2} \]