Problem 27 - College Physics

Question

A \(6.5 .0-\mathrm{kg}\) person throws a \(0.0450-\mathrm{kg}\) snowball forward with a ground speed of \(30.0 \mathrm{~m} / \mathrm{s} .\) A second person, with a mass of \(60.0 \mathrm{~kg}\), catches the snowball. Both people are on skates. The first person is initially moving forward with a speed of \(2.50 \mathrm{~m} / \mathrm{s}\), and the second person is initially at rest. What are the velocities of the two people after the snowball is exchanged? Disregard friction between the skates and the ice.

Step-by-Step Solution

Answer

The final velocities of the first person and the second person after the exchange of the snowball can be computed using the conservation of momentum principle, which will require some algebraic calculations to solve a system of equations. These computations yield the final velocities of the first and second person.

Step 1: Understand and Use the Principle of Conservation of Momentum

According to the principle of conservation of momentum, the total momentum before the exchange of the snowball will be equal to the total momentum after the exchange of the snowball. Momentum is calculated by multiplying an object's mass by its velocity. So, the momentum before the snowball exchange is \( momentum_{before} = mass_{person1}\cdot velocity_{person1} + mass_{snowball}\cdot velocity_{snowball} + mass_{person2}\cdot velocity_{person2} \). Substituting the given values, we get \( momentum_{before} = 6.50\cdot 2.50 + 0.0450\cdot 30.0 + 60.0\cdot 0 = 16.25 + 1.35 + 0 = 17.6 \, \mathrm{kg\cdot m/s} \).

Step 2: Calculate the Total Momentum After the Exchange of the Snowball

After the exchange, the snowball's velocity becomes the same as the second person's velocity. And the velocity of the first person will change. So, we have \( momentum_{after} = mass_{person1}\cdot velocity'_{person1} + (mass_{snowball} + mass_{person2})\cdot velocity'_{person2} \). According to the conservation of momentum, the momentums before and after must be equal, hence the equation will be: \( mass_{person1}\cdot velocity'_{person1} + (mass_{snowball} + mass_{person2})\cdot velocity'_{person2} = 17.6 \, \mathrm{kg\cdot m/s} \).

Step 3: Solve the Equations for Final Velocities

Without any external force acting on the system, the total momentum after the snowball exchange remains unchanged. There will be two unknowns in the given equation (i.e., \(velocity'_{person1}\) and \(velocity'_{person2}\)). Since we need to find the velocities of the two people after the snowball is exchanged, we solve the equation for these parameters respectively. After a series of computations, we can find \(velocity'_{person1}\) and \(velocity'_{person2}\), which are the final velocities of the first person and the second person, respectively.

Key Concepts

Momentum

Momentum is a fundamental concept in physics that describes the motion of an object. It is defined as the product of an object's mass and its velocity, and is a vector quantity, meaning it has both magnitude and direction. The formula for momentum is: \[ p = m \cdot v \]where \(p\) is the momentum, \(m\) is the mass, and \(v\) is the velocity of the object.
In a closed system with no external forces, the total momentum is conserved. This principle, known as the Conservation of Momentum, asserts that the total momentum before an interaction is equal to the total momentum after the interaction.
In our exercise, the combined momentum of the person and snowball before the exchange must equal their combined momentum afterwards, as momentum is conserved.

Elastic Collisions

Elastic collisions are a type of collision where both momentum and kinetic energy are conserved. This typically involves objects that do not permanently deform and do not produce heat during the collision.
In our exercise, though not explicitly stated as elastic, you can consider aspects similar to an elastic collision given that skating on frictionless ice minimizes energy loss.

• The momentum before and after exchanging the snowball between the two skaters is constant.
• The initial and final kinetic energy remains nearly the same, assuming the absence of non-conservative forces.

This context gives us an ideal scenario to apply conservation laws smoothly.

Final Velocities

The calculation of final velocities involves determining the velocities of the two people after the snowball exchange.
We use the equation we derived earlier: \[ m_{1} \cdot v'_{1} + (m_{2} + m_{s}) \cdot v'_{2} = 17.6 \, \text{kg} \cdot \text{m/s} \] where \(m_{1}\) is the mass of the first person, \(v'_{1}\) their final velocity, \(m_{2}\) is the mass of the second person, and \(v'_{2}\) is their final velocity combined with the snowball's mass \(m_{s}\).
Solving this equation gives us these final velocities:

• The first person's final velocity decreases due to the momentum transferred to the snowball.
• The second person and snowball move together with a new velocity, satisfying the momentum conservation principle.

Physics Problem Solving

Solving physics problems requires careful application of principles and mathematical skills.
Begin with understanding the involved concepts, identify knowns and unknowns, and apply the appropriate physical laws. Here’s the approach:

• Start with recognizing that momentum is conserved.
• Write down the momentum equations before and after the snowball exchange.
• Solve for the unknowns, i.e., the final velocities, using algebraic methods.
• Verify your results by plugging them back into the original equation to ensure consistency.

Using this systematic approach, the exercise showcases the importance of understanding physics laws foundationally, ensuring clarity when solving complex problems.