A block of mass m moving at a speed v compresses a spring through a distance x before its speed is halved. The spring constant of the spring is `(6mv^(2))/(nx^(2))`. Find value of n.

Answer

To solve the problem, we will use the principle of conservation of energy. The kinetic energy lost by the block is equal to the potential energy stored in the spring when it is compressed.

### Step-by-Step Solution:

1. **Initial Kinetic Energy of the Block:**
The initial kinetic energy (KE_initial) of the block moving at speed \( v \) is given by:
\[ KE_{\text{initial}} = \frac{1}{2} mv^2 \]

2. **Final Kinetic Energy of the Block:**
After the block compresses the spring and its speed is halved, its final speed is \( \frac{v}{2} \). The final kinetic energy (KE_final) 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. **Change in Kinetic Energy:**
The change in kinetic energy (ΔKE) as the block compresses the spring is:
\[ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} = \frac{1}{2} mv^2 - \frac{1}{8} mv^2 \]
To simplify this, we can find a common denominator (8):
\[ \Delta KE = \frac{4}{8} mv^2 - \frac{1}{8} mv^2 = \frac{3}{8} mv^2 \]

4. **Potential Energy Stored in the Spring:**
The potential energy (PE) stored in the spring when it is compressed by a distance \( x \) is given by:
\[ PE = \frac{1}{2} k x^2 \]
According to the conservation of energy, the change in kinetic energy is equal to the potential energy stored in the spring:
\[ \Delta KE = PE \implies \frac{3}{8} mv^2 = \frac{1}{2} k x^2 \]

5. **Solving for Spring Constant \( k \):**
Rearranging the equation to solve for \( k \):
\[ k = \frac{3 mv^2}{4 x^2} \]

6. **Given Spring Constant:**
We are given that the spring constant \( k \) can also be expressed as:
\[ k = \frac{6 mv^2}{n x^2} \]

7. **Setting the Two Expressions for \( k \) Equal:**
Now we can set the two expressions for \( k \) equal to each other:
\[ \frac{3 mv^2}{4 x^2} = \frac{6 mv^2}{n x^2} \]

8. **Cancelling Common Terms:**
We can cancel \( mv^2 \) and \( x^2 \) from both sides (assuming \( m \), \( v \), and \( x \) are not zero):
\[ \frac{3}{4} = \frac{6}{n} \]

9. **Cross-Multiplying to Solve for \( n \):**
Cross-multiplying gives:
\[ 3n = 24 \implies n = \frac{24}{3} = 8 \]

### Final Answer:
The value of \( n \) is \( 8 \).

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