A 0.5 kg block moving at a speed of `12 ms^( -1)` compresses a spring through a distance 30 cm when its speed is halved. The spring constant of the spring will be `Nm^(-1)`

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 after it has compressed the spring and the potential energy stored in the spring.

### Step-by-Step Solution:

1. **Identify the initial conditions:**
- Mass of the block, \( m = 0.5 \, \text{kg} \)
- Initial speed of the block, \( U = 12 \, \text{m/s} \)
- Distance compressed by the spring, \( x = 30 \, \text{cm} = 0.3 \, \text{m} \)

2. **Calculate the initial kinetic energy (KE_initial):**
\[ KE_{\text{initial}} = \frac{1}{2} m U^2 = \frac{1}{2} \times 0.5 \times (12)^2 \]
\[ KE_{\text{initial}} = \frac{1}{2} \times 0.5 \times 144 = 36 \, \text{J} \]

3. **Determine the speed of the block after compression:**
- The speed of the block after compression is halved:
\[ V = \frac{U}{2} = \frac{12}{2} = 6 \, \text{m/s} \]

4. **Calculate the kinetic energy after compression (KE_final):**
\[ KE_{\text{final}} = \frac{1}{2} m V^2 = \frac{1}{2} \times 0.5 \times (6)^2 \]
\[ KE_{\text{final}} = \frac{1}{2} \times 0.5 \times 36 = 9 \, \text{J} \]

5. **Calculate the potential energy stored in the spring (PE_spring):**
- The potential energy stored in the spring when compressed is given by:
\[ PE_{\text{spring}} = KE_{\text{initial}} - KE_{\text{final}} \]
\[ PE_{\text{spring}} = 36 \, \text{J} - 9 \, \text{J} = 27 \, \text{J} \]

6. **Relate the potential energy to the spring constant (k):**
- The potential energy stored in the spring is also given by:
\[ PE_{\text{spring}} = \frac{1}{2} k x^2 \]
- Setting the two expressions for potential energy equal gives:
\[ 27 = \frac{1}{2} k (0.3)^2 \]
\[ 27 = \frac{1}{2} k (0.09) \]
\[ 27 = 0.045 k \]

7. **Solve for the spring constant (k):**
\[ k = \frac{27}{0.045} = 600 \, \text{N/m} \]

### Final Answer:
The spring constant \( k \) is \( 600 \, \text{N/m} \).