**Problem 2:** A 4.0 kg block is moving at \( v_0 = 8 \text{ m/s} \) along a frictionless, horizontal surface towards a spring with force constant \( k = 400 \text{ N/m} \) that is attached to a wall. See Figure. The spring has negligible mass. (a) Find the maximum distance the spring will be compressed.(b) If the spring is to compress by no more than 0.15 m, what should be the maximum value of \( v_0 \)?### Explanation:This problem involves the concept of energy conservation where the initial kinetic energy of the block is converted into the potential energy of the compressed spring. 1. **Kinetic Energy**: Initial kinetic energy of the block is given by: \[ KE_{\text{initial}} = \frac{1}{2} m v_0^2 \] where \( m = 4.0 \text{ kg} \) and \( v_0 = 8 \text{ m/s} \).2. **Potential Energy of Spring**: The potential energy stored in a compressed spring is given by: \[ PE_{\text{spring}} = \frac{1}{2} k x^2 \] where \( k = 400 \text{ N/m} \) and \( x \) is the compression of the spring.3. **Maximum Compression**: Setting the initial kinetic energy equal to the potential energy of the spring at maximum compression: \[ \frac{1}{2} m v_0^2 = \frac{1}{2} k x^2 \] 4. **Solving Part (a)**: Isolate \( x \) to find the maximum compression: \[ x = \sqrt{\frac{m v_0^2}{k}} \] 5. **Solving Part (b)**: Given the maximum compression \( x = 0.15 \text{ m} \), solve for \( v_0 \) by rearranging the energy conservation equation: \[ v_0 = \sqrt{\frac{k x^2}{m}} \](Note: Ensure according to specific instructions in your educational material to add numerical solutions and final answers as per calculation if required.)

Name: **Problem 2:** A 4.0 kg block is moving at \( v_0 = 8 \text{ m/s} \)
Uploaded: 2024-03-20T06:58:04.000Z
Channel: Bartleby
Description: **Problem 2:** A 4.0 kg block is moving at \( v_0 = 8 \text{ m/s} \) along a frictionless, horizontal surface towards a spring with force constant \( k = 400 \text{ N/m} \) that is attached to a wall. See Figure. The spring has negligible mass. (a)