Question

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

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

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 (\(v_p\)). $$ p_{initial} = m_p \times v_p $$

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, $$ p_{final} = (m_p + m_c) \times v_f $$

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 \(v_f\): $$ v_f = \frac{m_p \times v_p}{m_p + m_c} $$

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: $$ KE_{initial} = \frac{1}{2}(m_p + m_c)v_f^2 $$

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 \(F_f\): $$ F_f = \frac{(m_p + m_c)v_f^2}{2d} $$

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.

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.

Key concepts

Momentum Conservation

Momentum conservation is a crucial principle in physics, especially when analyzing collisions. It tells us that in a closed system, the total momentum before an event is the same as the total momentum after. This is only accurate if no external forces affect the system.

In our exercise, the boy fires a pellet at a piece of cheese. The initial momentum is given entirely by the moving pellet. Since the cheese is initially at rest, its momentum starts at zero. By combining the momentum of both the pellet and the cheese after they stick together, we ensure the conservation of momentum.

To calculate this, we use the formula:- Initial momentum: \( p_{initial} = m_p \times v_p \)- Final momentum: \( p_{final} = (m_p + m_c) \times v_f \)By equating these, we solve for the final velocity, which is needed to understand how the energy is further transformed or dissipated due to friction.

Work-Energy Theorem

The work-energy theorem connects the work done by forces on an object to its change in kinetic energy. It states that the work done is equivalent to the change in kinetic energy. This theorem simplifies analyzing situations where forces cause motion.

In our scenario, after the cheese and pellet move together, they slide across the ice. Because they come to rest, this means all kinetic energy has been lost due to friction. Thus, the work done by friction equals the initial kinetic energy of the system.
Here's how this is represented mathematically:

• Work done by friction \( W = F_f \times d \)
• Change in kinetic energy \( \Delta KE = KE_{final} - KE_{initial} \)

Since the object stops,
- \( KE_{final} = 0 \)
- The friction converts kinetic energy into heat, stopping the cheese-pellet combo.

Kinetic Energy

Kinetic energy is the energy of motion. Any object moving with a velocity has kinetic energy, calculated using the formula: \( KE = \frac{1}{2}mv^2 \).
In our exercise, the combined mass of the pellet and cheese gives us the system's initial kinetic energy once it begins to slide on the ice. This kinetic energy is crucial as it determines how far they slide before coming to a stop.

The steps to finding this involve:- Calculating \( v_f \) using momentum conservation.- Substituting \( v_f \) back into the kinetic energy formula.
This calculated kinetic energy is then entirely dissipated by the work done by the frictional force, reflecting the physics principle of energy conversion.

Physics Problem Solving

When solving physics problems, especially involving collisions and motion, it is a multi-step process that requires careful application of principles like momentum conservation and the work-energy theorem.

Start by identifying the given quantities and what principles apply. In this exercise, knowing the masses, distances, and velocity is crucial.

• Apply momentum conservation to find unknown velocities.
• Use the work-energy theorem to relate kinetic energy and work done by forces like friction.
• Calculate associated forces, such as the frictional force and normal force.
• Finally, deduce other properties, like the coefficient of friction, which depend on these forces.

Following a structured approach ensures a systematic solution, reinforcing a deep understanding of physical concepts.