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Chapter 7: Problem 80

The principle of conservation of momentum is useful in some situations and not in others. Describe how you obtain the impulse-momentum theorem from Newton's Second Law and what situations lead to momentum conservation. How would you decide whether conservation of momentum could be useful in a particular problem?

Short Answer

Expert verified
Derive impulse-momentum theorem from \( F = ma \) and apply when net external force is zero. Use in isolated systems or collisions.

Step by step solution

01

- Understand Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration: \(F = ma\).
02

- Express Acceleration

Acceleration can be expressed as the change in velocity (\( \frac{{\triangle v}}{{\triangle t}} \)) over time. Therefore, \( F = m \frac{{\triangle v}}{{\triangle t}} \).
03

- Introduce Momentum and Impulse

Momentum (\( p \)) is defined as the product of mass and velocity (\( p = mv \)). Impulse (\( J \)) is defined as the product of force and time (\( J = F\triangle t \)).
04

- Substitute Momentum into Newton's Second Law

Since \( p = mv \), the change in momentum \( \triangle p \) can be written as \( m \triangle v \.\) Now, using Newton's Second Law: \( F = \frac{{\triangle p}}{{\triangle t}} \).
05

- Derive Impulse-Momentum Theorem

Rearrange \( F = \frac{{\triangle p}}{{\triangle t}} \) to \( F \triangle t = \triangle p \). This is known as the Impulse-Momentum Theorem.
06

- Identify Momentum Conservation Situations

Momentum is conserved when the net external force acting on a system is zero. This typically occurs in isolated systems or in collisions where external forces cancel out.
07

- Deciding Momentum Conservation Usefulness

Determine if external forces can be ignored or if they cancel out. In such cases, conservation of momentum proves useful to simplifying the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Second Law
Newton's Second Law of Motion is a fundamental principle in physics. It explains how the motion of an object changes when acted upon by a force. The law is famously written as: \( F = ma \) This formula states that the force (\

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Most popular questions from this chapter

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