System of particles, Linear Momentum


Objective

As a result of this lesson you will be able to

  1. Define the concept of momentum
  2. Describe the condition which the total momentum of a system of particles is constant
  3. Identify momentum as a separately conserved quantity different from energy
  4. Solve problems involving collisions
  5. Calculate momentum as the product of mass and velocity
  6. Discriminate between elastic, inelastic, and completely inelastic collisions
  7. Calculate the center of mass of a system
  8. Analyze situations involving a change in the mass of a body

Prerequisites

This topic expects you to have an understanding of the following in order to successfully learn all the material.

    • Newton's Forces
    • Conservation of Energy
    • Work-Energy Theorem
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Question

Would it be more dangerous to be hit by a small fast moving object or an object that is twice the mass and moving at half the speed?

Introduction

There are many situations in which we need to look at large systems of particles or determine what will occur when objects collide. This section will help to develop a new approach to solving problems dealing with collisions where we know very little about the forces involved. When ever we are unable to directly apply Newton's second law we will use the concepts of momentum, impulse, and the conservation of momentum. By using these new concepts we actually do not need to know anything about the forces involved. Now we can solve more complex situations such as collisions between multiple objects.
 

Connection

The concepts of momentum and impulse are derived from the idea of Newton’s second law. Instead of looking at acceleration in Newton’s law(\Sigma\overrightarrow{\textbf{F}}=m\overrightarrow{\textbf{a}}), we view it as the change in velocity over time(\overrightarrow{\textbf{a}}=\dfrac{d\overrightarrow{\textbf{v}}}{dt}) since acceleration is the rate of change(\dfract{rise}{run}) of velocity. When we do this we see that Newton’s second law is no longer mass times acceleration but is instead described in terms of a mass and a changing velocity. If you group mass and velocity together we get a concept called momentum. Now Force is the change of this new concept called momentum. Do not worry too much now if you dont see a clear connection because we will develop the relationship through this lesson.
 

Role

When we are unable to know the details about forces involved in a collision we can use the techniques of momentum and impulse to help us understand the situation. For example, we may wish to know which direction two cars move after they collide with each other. It may be difficult to determine the forces involved but we can easily determine information about the velocity, mass and the time over which the forces were applied. Whenever problems deal with collisions it is easy to use techniques that involve momentum and impulse to understand the problem.



Linear Momentum

 
Linear momentum is a vector quantity (direction and magnitude) that is in the same direction as velocity. Momentum is thought of as the quantity of motion and is the product of an object's mass and velocity. Since any object that we experience normally has mass, then any object that is in motion has momentum. Momentum is denoted as \overrightarrow{\textbf{p}}. It is useful to deal with momentum instead of the two quantities that it is made up of. Fow now you need to know that we are redefining Newton's second law in terms of momentum. This allows us to not have to worry about acceleration for now and gives us a new perspective when dealing with many problems.
Taking another look at Newton's Law

 

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Equation/Definition

The product of a particle's mass and velocity is called the momentum or linear momentum of the particle. Momentum is a vector so it has a magnitude and direction.

    \[\overrightarrow{\textbf{p}}=m\overrightarrow{\textbf{v}}\]


 

Example

What is the magnitude of the momentum of an object that has a velocity of 20\dfrac{m}{s^{2}} and a mass of 2,000kg? What would the velocity have to be in order for an object to maintain the same momentum and have a quarter of the mass?

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Impulse

 
Kinetic energy and momentum are closely related because they both depend on mass and velocity. The concept of impulse is used to help understand the difference between the two concepts. When we look at our new perspective of force and examine its change over a small period of time we develop a new concept called Impulse and is labeled as \overrightarrow{\textbf{J}}. Impulse is a vector quantity and it is the change in momentum over time. So why use this new concept? Impulse is the change in momentum over an interval AND the product of the net force and the same time interval.

Equation/Definition

The impulse of the net force is the product of the net force and the time interval. It is the force times the time interval. It is also the change in momentum over a time interval.
 
\overrightarrow{\textbf{J}}=\Sigma\overrightarrow{\textbf{F}}\Delta t=\overrightarrow{\textbf{p}}_2-\overrightarrow{\textbf{p}}_1


 

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Conservation of Linear Momentum

 
We use the concept of momentum when two objects interact with each other. Just as we saw with energy, in many situations it is conserved(meaning that it cannot be destroyed). This means the we can now look at an isolated system Take a look at the mathematical derivation below regarding the conservation of momentum.

Equation/Definition

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One important advantage of momentum is that it is conserved. We have been mentioning that momentum is used when objects collide. Well lets take a look at the forces and momentum of a simple linear collision. If two objects A and B collide then we will sum the forces that are experienced and use our new definition of force(in terms of momentum) to understand what is happening.
\overrightarrow{\textbf{F}}_{A\;on\;B} +\overrightarrow{\textbf{F}}_{B\;on\;A}=\dfrac{d\overrightarrow{\textbf{p}}_{B}}{dt}+\dfrac{d\overrightarrow{\textbf{p}}_{A}}{dt}= \dfrac{d(\overrightarrow{\textbf{p}}_{A}+\overrightarrow{\textbf{p}}_{B})}{dt}=0
Since the sum of forces is equal to zero, then the total change of momentum is zero. But what exactly does that mean? Well, it means that the momentum is constant. If we were to graph it we would see a straight horziontal line with a slope of zero for momenum.

Conservation of Momentum

 

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Collisions

 
Collisions occur when one object collides with another. This can occur in a few ways. Two object can collide and then bounce off each other or they can collide and become stuck and move together. We will now need two terms to describe the terms of collision.

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Center of Mass

 
Introduction description of topic or step

Equation/Definition

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