College Physics
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![Title: Kinetic Energy and Spring Compression Problem
**Problem Statement:**
A 0.5-kg block slides along a horizontal frictionless surface at 2 m/s. It is brought to rest by compressing a very long spring of spring constant 800 N/m. The maximum spring compression is:
- a) 0 cm
- b) 3 cm
- c) 5 cm
- d) 80 cm
- e) 80 m
**Solution Discussion:**
To find the maximum compression of the spring, we need to equate the initial kinetic energy of the block to the potential energy stored in the spring at maximum compression.
1. **Initial Kinetic Energy (KE):**
\[
KE = \frac{1}{2} m v^2
\]
where \( m = 0.5 \, \text{kg} \) and \( v = 2 \, \text{m/s} \).
2. **Potential Energy in the Spring (PE):**
\[
PE = \frac{1}{2} k x^2
\]
where \( k = 800 \, \text{N/m} \) and \( x \) is the compression.
3. **Setting KE equal to PE:**
\[
\frac{1}{2} m v^2 = \frac{1}{2} k x^2
\]
4. **Solving for \( x \):**
\[
x = \sqrt{\frac{m v^2}{k}}
\]
5. **Calculate:**
\[
x = \sqrt{\frac{0.5 \times 2^2}{800}} = \sqrt{\frac{2}{800}} = \sqrt{0.0025} = 0.05 \, \text{m}
\]
**Conclusion:**
The maximum spring compression is 0.05 m, which corresponds to option c) 5 cm.](https://content.bartleby.com/qna-images/question/500c0e80-687a-4e1b-8f8e-54233bb713dd/2d78ffb9-75d7-4241-9eb6-5acb40259208/wvonar_processed.png)
Transcribed Image Text:Title: Kinetic Energy and Spring Compression Problem
**Problem Statement:**
A 0.5-kg block slides along a horizontal frictionless surface at 2 m/s. It is brought to rest by compressing a very long spring of spring constant 800 N/m. The maximum spring compression is:
- a) 0 cm
- b) 3 cm
- c) 5 cm
- d) 80 cm
- e) 80 m
**Solution Discussion:**
To find the maximum compression of the spring, we need to equate the initial kinetic energy of the block to the potential energy stored in the spring at maximum compression.
1. **Initial Kinetic Energy (KE):**
\[
KE = \frac{1}{2} m v^2
\]
where \( m = 0.5 \, \text{kg} \) and \( v = 2 \, \text{m/s} \).
2. **Potential Energy in the Spring (PE):**
\[
PE = \frac{1}{2} k x^2
\]
where \( k = 800 \, \text{N/m} \) and \( x \) is the compression.
3. **Setting KE equal to PE:**
\[
\frac{1}{2} m v^2 = \frac{1}{2} k x^2
\]
4. **Solving for \( x \):**
\[
x = \sqrt{\frac{m v^2}{k}}
\]
5. **Calculate:**
\[
x = \sqrt{\frac{0.5 \times 2^2}{800}} = \sqrt{\frac{2}{800}} = \sqrt{0.0025} = 0.05 \, \text{m}
\]
**Conclusion:**
The maximum spring compression is 0.05 m, which corresponds to option c) 5 cm.
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