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

Explanation

To solve the problem, we will use the principle of conservation of energy. The initial kinetic energy of the block will be converted into the kinetic energy of the block when its speed is halved and the potential energy stored in the spring when it is compressed.

### Step-by-Step Solution:

1. **Identify Initial and Final Energies:**
- The initial kinetic energy (KE_initial) of the block when it is moving at speed \( v \) is given by:
\[ KE_{\text{initial}} = \frac{1}{2} mv^2 \]

2. **Determine Final Speed and Kinetic Energy:**
- When the block compresses the spring by a distance \( x \), its speed is halved. Thus, the final speed \( v_f \) is:
\[ v_f = \frac{v}{2} \]
- The final kinetic energy (KE_final) when the speed is halved is:
\[ KE_{\text{final}} = \frac{1}{2} m \left(\frac{v}{2}\right)^2 = \frac{1}{2} m \frac{v^2}{4} = \frac{1}{8} mv^2 \]

3. **Calculate Potential Energy in the Spring:**
- The potential energy (PE) stored in the spring when it is compressed by distance \( x \) is given by:
\[ PE = \frac{1}{2} k x^2 \]

4. **Apply Conservation of Energy:**
- According to the conservation of energy, the initial kinetic energy is equal to the sum of the final kinetic energy and the potential energy stored in the spring:
\[ KE_{\text{initial}} = KE_{\text{final}} + PE \]
- Substituting the expressions we derived:
\[ \frac{1}{2} mv^2 = \frac{1}{8} mv^2 + \frac{1}{2} k x^2 \]

5. **Rearranging the Equation:**
- To isolate the spring constant \( k \), we rearrange the equation:
\[ \frac{1}{2} mv^2 - \frac{1}{8} mv^2 = \frac{1}{2} k x^2 \]
- Finding a common denominator for the left side:
\[ \frac{4}{8} mv^2 - \frac{1}{8} mv^2 = \frac{3}{8} mv^2 \]
- Thus, we have:
\[ \frac{3}{8} mv^2 = \frac{1}{2} k x^2 \]

6. **Solving for the Spring Constant \( k \):**
- Multiply both sides by 2 to eliminate the fraction:
\[ \frac{3}{4} mv^2 = k x^2 \]
- Finally, solve for \( k \):
\[ k = \frac{3mv^2}{4x^2} \]

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