Problem 32 - College Physics

Question

\(\bullet\) On a cold winter day, a penny (mass 2.50 g) and a nickel (mass 5.00 g) are lying on the smooth (frictionless) surface of a frozen lake. With your finger, you flick the penny toward the nickel with a speed of 2.20 \(\mathrm{m} / \mathrm{s}\) . The coins collide elastically; calculate both their final velocities (speed and direction).

Step-by-Step Solution

Answer

The penny's final velocity is 0.60 m/s, and the nickel's final velocity is 2.80 m/s, both moving in the same direction.

Step 1: Analyze the Problem

Two masses are involved: a penny with mass \( m_1 = 2.50 \text{ g} = 0.0025 \text{ kg} \) and a nickel with mass \( m_2 = 5.00 \text{ g} = 0.005 \text{ kg} \). The initial velocity of the penny \( v_1 = 2.20 \text{ m/s} \) and the nickel is stationary with \( v_2 = 0 \text{ m/s} \). The collision is elastic, meaning both momentum and kinetic energy are conserved.

Step 2: Apply Conservation of Momentum

The law of conservation of momentum in a closed system states \( m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2' \), where \( v_1' \) and \( v_2' \) are the final velocities. Substituting the given values: \[ 0.0025 \times 2.20 + 0.005 \times 0 = 0.0025 v_1' + 0.005 v_2' \]\[ 0.0055 = 0.0025 v_1' + 0.005 v_2' \]

Step 3: Apply Conservation of Kinetic Energy

For an elastic collision, kinetic energy is also conserved: \[ \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2 \]Simplifying with the known values: \[ \frac{1}{2} \times 0.0025 \times (2.20)^2 = \frac{1}{2} \times 0.0025 \times v_1'^2 + \frac{1}{2} \times 0.005 \times v_2'^2 \]\[ 0.00605 = 0.00125 v_1'^2 + 0.0025 v_2'^2 \]

Step 4: Solve the System of Equations

We now solve the equations from Steps 2 and 3 simultaneously: From momentum:\[ 0.0055 = 0.0025 v_1' + 0.005 v_2' \] and From energy: \[ 0.00605 = 0.00125 v_1'^2 + 0.0025 v_2'^2 \].Solving these using algebraic methods or a calculator gives \( v_1' = 0.60 \text{ m/s} \) and \( v_2' = 2.80 \text{ m/s} \).

Step 5: Interpret Final Velocities

After solving, the penny's final velocity \( v_1' \) is \( 0.60 \text{ m/s} \) (slowing down), and the nickel's final velocity \( v_2' \) is \( 2.80 \text{ m/s} \) (moving in the same direction the penny was initially moving).

Key Concepts

Conservation of Momentum

In the context of an elastic collision, conservation of momentum is an exciting concept. Imagine two ice skaters pushing off each other on a smooth, slippery surface. The penny and nickel in our problem are quite similar. They're initially motionless on a frictionless surface, but once they're flicked, they interact and exchange momentum.
The law of conservation of momentum tells us that the total momentum before the collision must be equal to the total momentum afterward. For this exercise:

• The penny travels at 2.20 m/s, contributing its part to the initial momentum.
• The nickel starts stationary, so its initial momentum is zero.

When they collide, this conserved momentum distributes between them. Our equation becomes: \[ m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2' \] where v_1' and v_2' are the velocities post-collision. It's as if they're sharing the kinetic energy and momentum equally, although the nickel will end up with a larger share due to its greater mass in this interaction.

Conservation of Kinetic Energy

Just like momentum, kinetic energy is another vital aspect of an elastic collision. But unlike the straightforward transfer of momentum, energy has a cool twist. For kinetic energy to remain conserved, the energy type involved in the collision is merely transferred--not lost.
For the penny-nickel collision, the principle of conservation of kinetic energy dictates that:

• The sum of the kinetic energies before the collision must equal the sum after.
• The formula involving squares helps capture how velocity influences energy more than momentum.

Using this, we write the equation for conservation of kinetic energy as: \[ \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2 \] The fact that both energy and momentum need conserving might seem a bit intricate at first. But with both remaining steady, it ensures that nothing is lost to sound, heat, or other modes of energy--making the penny and nickel as efficient as possible in this bouncing scenario.

Final Velocities Calculation

After understanding the key principles, the actual math of finding the final velocities might feel like detective work. You'll have two ratios--one from momentum and one from energy--to play with.
Given the prior two equations, solving them will involve some basic algebra: arranging, substituting, and simplifying variables.

• Starting with our momentum equation, it ensures that total forces felt by each puck balance out.
• Energy conservation adds detail about how quickly each can move while adhering to physical laws.

Once you work through the math, you'll find:

• The penny comes to rest at 0.60 m/s, moving slower than before.
• The nickel, now sporting more of the leftover energy, moves at 2.80 m/s.

The beauty lies in the math's predictive power, showing how such universal laws lead to these accessible, yet fascinating calculations! Every number has meaning, expertly peering into the results of these coalescing dynamics.