College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![### Problem Statement
A bullet with a mass \( m_b = 11.5 \, \text{g} \) is fired into a block of wood at a velocity \( v_b = 257 \, \text{m/s} \). The block is attached to a spring with a spring constant \( k = 205 \, \text{N/m} \). The block and bullet continue to move, compressing the spring by 35.0 cm before the whole system momentarily comes to a stop. Assuming that the surface on which the block is resting is frictionless, determine the mass \( m_w \) of the wooden block.
\[ m_w = \, \text{________ kg} \]
### Diagram Explanation
To the right of the text, there is a diagram illustrating the scenario:
- A bullet, labeled with \( m_b \) and moving with velocity \( v_b \), is shown approaching a block in motion.
- The block is depicted as a red rectangle connected to a horizontal spring, which is attached to a fixed support.
- The spring is shown in a compressed state, indicating the final momentary stop of the system.
### Assumptions and Analysis
1. **Conservation of Momentum**:
- When the bullet embeds into the block, linear momentum is conserved.
- Initial momentum = Final momentum
2. **Energy Conservation**:
- The kinetic energy of the bullet-block system transforms into potential energy of the spring.
- \( \frac{1}{2}k x^2 = \frac{1}{2}(m_b + m_w)v^2 \), where \( x = 0.35 \, \text{m} \).
This analysis allows us to solve for the unknown mass \( m_w \) of the wooden block.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9081bc6b-9817-477c-9b8d-7c2dcc8c93d8%2F16f1b685-77e5-48da-8711-0930d9f84cb9%2Fptz0hq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A bullet with a mass \( m_b = 11.5 \, \text{g} \) is fired into a block of wood at a velocity \( v_b = 257 \, \text{m/s} \). The block is attached to a spring with a spring constant \( k = 205 \, \text{N/m} \). The block and bullet continue to move, compressing the spring by 35.0 cm before the whole system momentarily comes to a stop. Assuming that the surface on which the block is resting is frictionless, determine the mass \( m_w \) of the wooden block.
\[ m_w = \, \text{________ kg} \]
### Diagram Explanation
To the right of the text, there is a diagram illustrating the scenario:
- A bullet, labeled with \( m_b \) and moving with velocity \( v_b \), is shown approaching a block in motion.
- The block is depicted as a red rectangle connected to a horizontal spring, which is attached to a fixed support.
- The spring is shown in a compressed state, indicating the final momentary stop of the system.
### Assumptions and Analysis
1. **Conservation of Momentum**:
- When the bullet embeds into the block, linear momentum is conserved.
- Initial momentum = Final momentum
2. **Energy Conservation**:
- The kinetic energy of the bullet-block system transforms into potential energy of the spring.
- \( \frac{1}{2}k x^2 = \frac{1}{2}(m_b + m_w)v^2 \), where \( x = 0.35 \, \text{m} \).
This analysis allows us to solve for the unknown mass \( m_w \) of the wooden block.
Expert Solution
Step 1: Given values
Given,
mw (to be computed) be the mass of the wooden block,
vb = 245 m/s be the velocity of the bullet,
v (unknown) be the initial velocity of the block with the bullet lodged in,
k = 205 N/m be the spring constant,
mb = 13.5 g = 0.0135 kg be the mass of the bullet,
s = 35 cm = 0.35 m be the compression of the spring.
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