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Chapter 2: Problem 23

Jane and John are both on roller skates and are facing each other. First Jane pushes John with her hands and they move apart. Later they get together, and John pushes Jane equally hard with his hands and they move apart. Do they move any differently in the two cases? Why or why not?

Short Answer

Expert verified
Answer: There is no difference in their motion in either case, as they move with the same velocities due to the conservation of momentum and the action-reaction forces.

Step by step solution

01

Understanding the Scenario:

In the first case, Jane pushes John, causing both of them to move apart. In the second case, John pushes Jane with an equal force, again causing them to move apart. We need to compare these two cases and understand if there is any difference in their motions.
02

Concepts and Principles:

We will be using the principle of conservation of momentum in order to solve this problem. The conservation of momentum states that in the absence of external forces, the total momentum of a system of particles remains constant.
03

Define the System:

In this case, our system consists of Jane and John. Since there are no external forces acting on the system (we can neglect air friction and rolling friction), the total momentum of the system should be conserved.
04

Calculating Initial Momentum:

We know that momentum is calculated by the product of mass and velocity (momentum = mass x velocity). Initially, before the push, both Jane and John are stationary, so their initial velocity is 0. Thus, the initial total momentum of the system is 0.
05

First Case: Jane Pushes John

When Jane pushes John, she applies a force on John (F) causing him to move with a certain velocity (v_j). According to Newton's Third Law of Motion, John also applies an equal and opposite force (-F) on Jane. This causes Jane to move with a certain velocity (v_a) in the opposite direction. In order to conserve the momentum, we can write the following equation: m_j * v_j = m_a * (-v_a)
06

Second Case: John Pushes Jane

When John pushes Jane, he applies a force on Jane (F), causing her to move with a certain velocity (v_a'). According to Newton's Third Law of Motion, Jane also applies an equal and opposite force (-F) on John. This causes John to move with a certain velocity (v_j') in the opposite direction. Similar to the first case, we can write the following equation to conserve momentum: m_j * (-v_j') = m_a * v_a'
07

Comparing the Two Cases

We can now compare the two equations: m_j * v_j = m_a * (-v_a) m_j * (-v_j') = m_a * v_a' The forces in both cases are equal and opposite, so we can conclude that v_j = v_j' and v_a = v_a', indicating that they move with the same velocities in both cases. Therefore, there is no difference in their motion in either case. In conclusion, Jane and John move with the same velocities in both cases due to the conservation of momentum and the action-reaction forces.

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Most popular questions from this chapter

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