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Chapter 7: Problem 88

A bored boy shoots a soft pellet from an air gun at a piece of cheese with mass \(0.25 \mathrm{~kg}\) that sits, keeping cool for dinner guests, on a block of ice. On one particular shot, his 1.2-g pellet gets stuck in the cheese, causing it to slide \(25 \mathrm{~cm}\) before coming to a stop. According to the package the gun came in, the muzzle velocity is \(65 \mathrm{~m} / \mathrm{s}\). What is the coefficient of friction between the cheese and the ice?

Short Answer

Expert verified
Question: Find the coefficient of friction between the cheese and the ice, given the mass of the cheese, the mass and muzzle velocity of the pellet, and the sliding distance of the cheese after getting hit by the pellet. Answer: To find the coefficient of friction between the cheese and the ice, you need to follow these steps: 1. Calculate the initial momentum of the system. 2. Calculate the final momentum of the system. 3. Use the conservation of momentum to find the final velocity of the cheese after the pellet hits it. 4. Use the work-energy theorem to find the work done in overcoming friction. 5. Calculate the friction force. 6. Find the normal force between the cheese and the ice. 7. Calculate the coefficient of friction by dividing the friction force by the normal force. Plug in the given values into the equations derived in each step and solve for the coefficient of friction.

Step by step solution

01

Calculate initial momentum of the system

The initial momentum of the system consists only of the momentum of the pellet. We have mass of pellet (\(m_p\)) and its muzzle velocity (\(v_p\)). $$ p_{initial} = m_p \times v_p $$
02

Calculate the final momentum of the system

In the final state, the cheese and the pellet move together. Let their final combined velocity be \(v_f\). The mass of the cheese is \(m_c\). Then, $$ p_{final} = (m_p + m_c) \times v_f $$
03

Use the conservation of momentum

Using the conservation of momentum, we can set the initial momentum equal to the final momentum. $$ m_p \times v_p = (m_p + m_c) \times v_f $$ Solve for \(v_f\): $$ v_f = \frac{m_p \times v_p}{m_p + m_c} $$
04

Use the work-energy theorem

The work done by friction during the sliding motion is equal to the change in kinetic energy of the cheese-pellet system. Let \(F_f\) be the friction force and \(d\) be the sliding distance. Then, the work done by friction force is: $$ W = F_f \times d $$ The work-energy theorem states that: $$ W = \Delta KE = KE_{final} - KE_{initial} $$ Since the system comes to a stop, \(KE_{final} = 0\). The initial kinetic energy is: $$ KE_{initial} = \frac{1}{2}(m_p + m_c)v_f^2 $$
05

Calculate the friction force

From the previous step, we have: $$ F_f \times d = \frac{1}{2}(m_p + m_c)v_f^2 $$ Solve for \(F_f\): $$ F_f = \frac{(m_p + m_c)v_f^2}{2d} $$
06

Find the normal force

The normal force (\(F_N\)) is equal to the weight of the cheese and the pellet. $$ F_N = g(m_p + m_c) $$ where \(g = 9.8 \mathrm{~m/s^2}\) is the gravitational acceleration.
07

Calculate the coefficient of friction

The coefficient of friction (\(\mu\)) can be found using the friction force and normal force: $$ \mu = \frac{F_f}{F_N} $$ Substitute the values from Step 5 and Step 6, and solve for \(\mu\). After calculating the values, you will get the coefficient of friction between the cheese and the ice.

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Most popular questions from this chapter

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