Elastic and Inelastic Collisions
Ck12 Science
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both in
the U.S. and worldwide. Using an open-source, collaborative, and
web-based compilation model, CK-12 pioneers and promotes the
creation and distribution of high-quality, adaptive online textbooks
that can be mixed, modified and printed (i.e., the FlexBook®
textbooks).
Copyright © 2015 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/about/
terms-of-use.
Printed: December 10, 2015
AUTHOR
Ck12 Science
www.ck12.org
C HAPTER
Chapter 1. Elastic and Inelastic Collisions
1
Elastic and Inelastic
Collisions
• Describe the difference between elastic and inelastic collisions and indicate what is conserved in each case.
• Solve problems involving elastic collisions using both the conservation of momentum and the conservation of
kinetic energy.
This device is known as Newton’s cradle. As the balls collide with each other, nearly all the momentum and kinetic
energy is conserved. If one ball swings down, exactly one ball will swing up; if three balls swing down, exactly
three will swing back up. The collisions between the balls are very nearly elastic.
Elastic and Inelastic Collisions
For all collisions in a closed system, momentum is conserved. In some collisions in a closed system, kinetic energy is
conserved. When both momentum and kinetic energy are conserved, the collision is called an elastic collision. Most
collisions are inelastic because some amount of kinetic energy is converted to potential energy, usually by raising
one of the objects higher (increasing gravitation PE) or by flexing the object. Any denting or other changing of
shape by one of the objects will also be accompanied by a loss of kinetic energy. The only commonly seen elastic
collisions are those between billiard balls or ball bearings, because these balls do not compress. And, of course,
collisions between molecules are elastic if no damage is done to the molecules.
Much more common are inelastic collisions. These collisions occur whenever kinetic energy is not conserved,
primarily when an object’s height is increased after the collision or when one of the objects is compressed.
Example Problem: A 12.0 kg toy train car moving at 2.40 m/s on a straight, level train track, collides head-on with
a second train car whose mass is 36.0 kg and was at rest on the track. If the collision is perfectly elastic and all
motion is frictionless, calculate the velocities of the two cars after the collision.
1
www.ck12.org
Solution: Since the collision is elastic, both momentum and KE are conserved. We use the conservation of momentum and conservation of KE equations.
Conservation of momentum: m1 v1 + m2 v2 = m1 v1 0 + m2 v2 0
Conservation of KE: 12 m1 v1 2 + 12 m2 v22 = 12 m1 v1 0 2 = 21 m2 v2 0 2
Since m1 , m2 , v1 , and v2 are known, only v1 0 and v2 0 are unknown. When the known values are plugged into these
two equations, we will have two equations with two unknowns. Such systems can be solved with algebra.
(12.0 kg)(2.40 m/s) + (36.0 kg)(0 m/s) = (12.0 kg)(v1 0 ) + (36.0 kg)(v2 0 )
28.8 = 12.0 v1 0 + 36.0 v2 0
Solving this equation for v1 0 yields v1 0 = 2.4 − 3 v2 0
1
1
2
2
2 (12.0)(2.40) + 2 (36.0)(0)
69.1 = 12.0 v1 0 2 + 36.0 v2 0 2
= 21 (12.0)(v1 0 )2 + 12 (36.0)(v2 0 )2
5.76 = v1 0 2 + 3 v2 0 2
Substituting the equation for v1 0 into this equation yields
5.76 = (2.4 − 3 v2 0 )2 + 3 v2 0 2
5.76 = 5.76 − 14.4 v2 0 + 9 v2 0 2 + 3 v2 0 2
12 v2 0 2 − 14.4 v2 0 = 0
12 v2 0 = 14.4
v2 0 = 1.2 m/s
Substituting this result back into v1 0 = 2.4 − 3 v2 0 , we get v1 0 = −1.2 m/s.
So, the heavier car is moving in the original direction at 1.2 m/s and the lighter car is moving backward at 1.2 m/s.
The Science Of Bouncing
What is bouncing? Why do some rubber balls bounce more than others? What does this have to do with energy
conversion? Find out at http://youtu.be/9MwyVpVdvVA?list=PLzMhsCgGKd1hoofiKuifwy6qRXZs7NG6a.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/143890
Summary
• Elastic collisions are those in which both momentum and kinetic energy are conserved.
• Inelastic collisions are those in which either momentum or kinetic energy is not conserved.
Practice
Questions
2
www.ck12.org
Chapter 1. Elastic and Inelastic Collisions
The following video is a demonstration of elastic and inelastic collisions. Use this resource to answer the questions
that follow.
http://www.teachertube.com/viewVideo.php?video_id=30870
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/82509
1. Explain what happened in the first demonstration on elastic collisions.
2. Explain what happened in the second demonstration on inelastic collisions.
3. Assuming the first carts started at the same speed in both demonstrations, explain using momentum why the
inelastic collision ended slower than the elastic collision.
Practice problems for elastic collision:
http://courses.ncssm.edu/aphys/problems/collisions/elascoll.html
Review
Questions
1. A 4.00 kg metal cart is sitting at rest on a frictionless ice surface. Another metal cart whose mass is 1.00 kg
is fired at the cart and strikes it in a one-dimensional elastic collision. If the original velocity of the second
cart was 2.00 m/s, what are the velocities of the two carts after the collision?
2. Identify the following collisions as most likely elastic or most likely inelastic.
1.
2.
3.
4.
5.
A ball of modeling clay dropped on the floor.
A fender-bender automobile collision.
A golf ball landing on the green.
Two billiard balls colliding on a billiard table.
A collision between two ball bearings.
• elastic collision: One in which both momentum and kinetic energy are conserved.
References
1. Flickr: hellolapomme. http://www.flickr.com/photos/hellolapomme/2289020306/ .
3