Problem 30 - College Physics

Question

On a cold winter day, a penny (mass \(2.50 \mathrm{~g}\) ) and a nickel (mass \(5.00 \mathrm{~g}\) ) 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 head-on elastically; calculate the final velocities (speed and direction) of both.

Step-by-Step Solution

Answer

Penny: 0.732 m/s opposite; Nickel: 1.47 m/s forward.

Step 1: Define the Known Variables

First, we list the known values: the mass of the penny \(m_1 = 2.50 \text{ g} = 0.0025 \text{ kg}\), the mass of the nickel \(m_2 = 5.00 \text{ g} = 0.005 \text{ kg}\), the initial velocity of the penny \(v_1 = 2.20 \text{ m/s}\), and the initial velocity of the nickel \(v_2 = 0 \text{ m/s}\).

Step 2: Apply Conservation of Momentum

In an elastic collision, both momentum and kinetic energy are conserved. Using the conservation of momentum, \(m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'\). Plugging in the given values:\[0.0025 \times 2.20 = 0.0025 \times v_1' + 0.005 \times v_2'\].

Step 3: Apply Conservation of Kinetic Energy

The kinetic energy is also conserved, given by the equation: \(\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2\). Substituting the known values:\[\frac{1}{2} \times 0.0025 \times (2.20)^2 = \frac{1}{2} \times 0.0025 \times {v_1'}^2 + \frac{1}{2} \times 0.005 \times {v_2'}^2\].

Step 4: Solve the System of Equations

We now solve the system of two equations with two variables \(v_1'\) and \(v_2'\). From momentum conservation, we have:\[0.0055 = 0.0025v_1' + 0.005v_2'\].And from kinetic energy conservation:\[0.00605 = 0.00125{v_1'}^2 + 0.0025{v_2'}^2\].By solving these equations simultaneously, we find \(v_1' = -0.732 \text{ m/s}\) and \(v_2' = 1.47 \text{ m/s}\).

Step 5: Interpret the Results

The negative sign of \(v_1'\) indicates that after the collision, the penny moves in the opposite direction of its initial movement, with a speed of \(0.732 \text{ m/s}\). The nickel moves in the same direction as the initial movement of the penny with a speed of \(1.47 \text{ m/s}\).

Key Concepts

Conservation of Momentum

In the world of physics, momentum plays a crucial role in understanding motion. Momentum is conserved when no external forces are acting on a closed system, as in the case of two colliding coins. The law of conservation of momentum states that the total momentum before the collision equals the total momentum after the collision.

To write this more formally, in our problem, the equation is:

• Before the collision: \( m_1v_1 + m_2v_2 \)
• After the collision: \( m_1v_1' + m_2v_2' \)

This equation helps us relate the initial and final velocities, because we know the initial condition. Understanding momentum is not only essential in elastic collisions but also in a variety of areas of physics. It provides a clear way to predict movement outcomes without delving into more complex dynamics immediately.

Conservation of Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. In an elastic collision, like the one between the penny and the nickel, kinetic energy is conserved. This means that the total kinetic energy of the system does not change during the collision.

The equation for conservation of kinetic energy is:

• Before the collision: \( \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \)
• After the collision: \( \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2 \)

Despite the interaction during the collision, the total kinetic energy remains constant.

This property of elastic collision ensures energy efficiency in the system, ultimately balancing work done before and after the interaction. By using both conservation of momentum and conservation of kinetic energy, we gain a powerful system of equations that allow us to solve for unknowns like final velocities.

Collision Dynamics

Collision dynamics involves analyzing the motion and interaction between colliding bodies and understanding how they exchange energy and momentum. When dealing with elastic collisions, we can predict how objects will move post-impact by using the conservation laws.

Elastic collisions can be characterized by:

• No loss of kinetic energy
• Momentum conservation
• Objects often bounce off each other

The penny and nickel example reveals that after they collide, the penny moves backward, while the nickel goes forward. This exchange showcases the principles of inversion and transfer in collision dynamics, where speed and direction transform while obeying conservation laws.

Problem Solving in Physics

Problem-solving in physics often involves identifying and applying theoretical concepts to tangible scenarios. By breaking down the problems into smaller steps, we can systematically solve complex equations. In our property collision problem, this approach included:

• Listing known and unknown variables
• Applying relevant conservation laws
• Setting up equations
• Solving these equations simultaneously

This logical approach is not only applicable to collision problems but serves as a foundational method in physics generally.

Ultimately, practice and understanding of underlying physics principles underpin successful problem-solving in physics, building valuable skills for analyzing various real-world situations and phenomena.