Problem 32 - University Physics with Modern Physics

Question

Two skaters collide and grab on to each other on frictionless ice. One of them, of mass \(70.0 \mathrm{kg},\) is moving to the right at \(2.00 \mathrm{m} / \mathrm{s},\) while the other, of mass \(65.0 \mathrm{kg},\) is moving to the left at 2.50 m/s. What are the magnitude and direction of the velocity of these skaters just after they collide?

Step-by-Step Solution

Answer

The skaters move to the left at a velocity of 0.167 m/s.

Step 1: Understand the Conservation of Momentum

In this problem, we need to use the principle of conservation of momentum. Momentum is conserved in isolated systems, which means the total momentum before the collision is equal to the total momentum after the collision. This principle is crucial as there is no external force applied here.

Step 2: Identify the Initial Momenta

First, calculate the initial momentum of each skater. The formula for momentum is given by \( p = mv \), where \( m \) is mass and \( v \) is velocity.- The initial momentum of the 70 kg skater: \( p_1 = 70 \, \text{kg} \, \times \, 2.00 \, \text{m/s} = 140 \, \text{kg m/s} \).- The initial momentum of the 65 kg skater: \( p_2 = 65 \, \text{kg} \, \times \, (-2.50) \, \text{m/s} = -162.5 \, \text{kg m/s} \) (the negative sign indicates direction).

Step 3: Calculate Total Initial Momentum

Add the two momenta to get the total initial momentum: \( P_{\text{total initial}} = 140 \, \text{kg m/s} + (-162.5 \, \text{kg m/s}) = -22.5 \, \text{kg m/s} \).

Step 4: Determine Total System Mass

The total mass of the system after the skaters collide is the sum of the two individual masses: \( M_{\text{total}} = 70 \, \text{kg} + 65 \, \text{kg} = 135 \, \text{kg} \).

Step 5: Apply Conservation of Momentum

According to the conservation of momentum, \( P_{\text{total initial}} = P_{\text{total final}} \). Since \( P_{\text{total final}} = M_{\text{total}} \times v_{\text{after}} \), we substitute this into the equation: \(-22.5 \, \text{kg m/s} = 135 \, \text{kg} \times v_{\text{after}} \).

Step 6: Solve for Final Velocity

Solve the equation for the velocity \( v_{\text{after}} \):\[ v_{\text{after}} = \frac{-22.5 \, \text{kg m/s}}{135 \, \text{kg}} = -0.167 \, \text{m/s} \].The negative sign indicates that the direction of the velocity is to the left.

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 velocity, usually expressed in the formula:

• \( p = m \cdot v \)
• Where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity of the object.

In our skater example, each skater has their own momentum due to their respective masses and velocities. The skater moving to the right has a positive momentum, while the skater moving to the left has a negative momentum due to the opposite direction of motion.
Momentum is vector-based, meaning the direction of the velocity affects the sign (positive or negative) of the momentum. By considering both direction and magnitude, we can accurately assess the motion in any collision scenario.

Elastic Collision

Elastic collisions are special types of collisions where the total kinetic energy and total momentum of the system are both conserved. Although the skaters in this example stick together post-collision, making it a perfectly inelastic collision,
understanding elastic collisions provides a base understanding of how momentum works in general.

If the skaters had rebounded off one another instead of sticking together, we would need to ensure that their total momentum both before and after the collision stayed the same, alongside their kinetic energy. This principle helps us better understand the mechanics behind momentum conservation in various scenarios.

System Mass

The system mass refers to the total combined mass of all objects involved in a situation. In the context of our skater problem, the system mass is simply the sum of the masses of both skaters:

• First skater's mass: 70 kg
• Second skater's mass: 65 kg
• Total system mass: \( 70 + 65 = 135 \) kg

Understanding the system mass is crucial when analyzing collisions, as it helps determine the overall dynamics after the collision. The larger the total mass, the smaller the impact of velocity changes on the system's total motion.
This plays a pivotal role in calculating the final velocity of the system based on momentum conservation.

Velocity Calculation

In momentum problems, velocity calculation often follows the application of the conservation of momentum principle. For the skaters, the velocity after the collision can be calculated using their combined momentum:

• Total initial momentum: \(-22.5 \) kg m/s
• Total system mass: 135 kg
• Formula: \( v_{\text{after}} = \frac{\text{Total Initial Momentum}}{\text{Total Mass}} \)

Using these values results in:\[v_{\text{after}} = \frac{-22.5 \, \text{kg m/s}}{135 \, \text{kg}} = -0.167 \, \text{m/s}\]The negative sign indicates that the direction of the velocity after collision is to the left, which matches the direction of the larger initial negative momentum of the system.
This calculation demonstrates the importance of understanding both direction and magnitude in evaluating post-collision conditions.