Unformatted text preview: Laboratory 5
OneDimensional Collisions
Introduction
In today’s laboratory, we introduce the concepts of work, energy, and energy conservation. Consider a system of objects (such as two gliders on a frictionless air track). When no outside work
is done on the system, the total energy of the system is “conserved”: it is constant. In this laboratory, we look at the concept of energy conservation in the context of onedimensional collisions
on a frictionless surface. We also investigate another conservation law, the conservation of linear
momentum, which can be derived directly from Newton’s Second Law. Theoretical Section: Work and Energy
The work done on an object by a constant force, , is deﬁned as
F
W = Fd cos(θ ),
where F is the magnitude of the force, d is the magnitude of the object’s displacement (as a result
of the action of the force), and θ is the angle between the force and displacement vectors. Work
has units of Joules, where 1 J = 1 N m. In vector notation, we write
W = · d,
F
and say that the work done is equal to the dot product of the force and displacement vectors.
Work is intimately related to energy; in fact, the term “energy” is deﬁned as the capacity of an
object to do work. Work is also deﬁned  somewhat circularly  as “the transfer of energy from one
system to another”. We all have an intuitive sense of what “energy” and “work” refer to; all that is
left is to describe them mathematically.
There are two major types of energy: potential energy, U , and kinetic energy, K . We can think
of potential energy as energy stored within a physical system: it has to do with the position of a
system, rather than its motion. For example, gravitational potential energy is stored in an object
when it is moved from a lower to a higher position relative to Earth’s surface. The change in
potential energy that occurs when an object is moved a vertical distance h is
∆Ug = mgh.
39 (1) Another form of potential energy is elastic potential energy, which is stored when one compresses
or stretches a spring away from its equilibrium position. The amount of energy stored when the
spring is stretched or compressed a distance x is
1
Ue = kx2 ,
2 (2) where k is the spring constant. Kinetic energy, in contrast, is associated with the motion of a
system. For a single object of mass m moving at a speed v, the kinetic energy is
1
K = mv2 .
2 (3) All of these formulas should be familiar from lecture.
Problem 5.1 What are the two basic types of energy?
Problem 5.2 What is an example of a situation where gravitational potential energy is converted into kinetic energy? How about a situation where
kinetic energy is converted into gravitational potential energy? Elastic potential energy to kinetic energy? Theoretical Background: Conservation of Energy
We have already seen two types of energy: kinetic and potential. These are really the only two
fundamental types of energy. However, most physics textbooks also deﬁne a third type: internal
energy. The internal energy of a body is deﬁned as the sum total of the kinetic energy of all
of its component molecules (translational, rotational, and vibrational) plus the potential energy
associated with the vibrational and electrostatic energy of atoms within these molecules. This
includes all of the energy in all of the chemical bonds in the object, as well as the energy of free
electrons ﬂoating around within it. We will abbreviate an object’s potential energy as U , its kinetic
energy as K , and its internal energy as N .
The Law of Conservation of Energy states that if no external work is done on a system, the sum
of its potential, kinetic, and internal energies must be constant in time. What does this mean? It
means that if the system gains one type of energy, it must lose some of another type of energy to
compensate. Mathematically, this is written as
∆K + ∆U + ∆N = 0. (4) What does this mean in the context of the current experiment, which is about collisions? We will
examine collisions where the gravitational potential energy of the colliding objects does not change
(i.e. they collide on a level surface), and there are no springs or other sources of elastic potential
energy involved, so the only two types of energy we still have to worry about are kinetic energy
40 and internal energy.
Physicists deﬁne two types of collisions: elastic collisions and inelastic collisions. The sole
difference between these two types is that in an elastic collision, the total kinetic energy of the two
colliding objects is the same before and after the collision. The objects simply transfer some of the
kinetic energy between themselves, and no kinetic energy is converted into internal energy (and
vice versa). In other words, kinetic energy is conserved in elastic collisions.
In inelastic collisions, however, kinetic energy is not conserved. Instead, some of it is transformed
into the internal energy of the colliding objects while the collision is taking place.
The difference is somewhat intuitive. An example of an inelastic collision would be one between
two balls made of putty  the objects are “squishy”, they sometimes deform during the collision,
and they stick together afterward. A series of snapshots of two inelastic collisions is shown in
Figure 1. An example of an elastic collision, in contrast, would be one between two pool balls the objects are hard, do not change shape during the collision, and bounce off each other cleanly.
A series of snapshots of two elastic collisions is shown in Figure 2. There are also many types of
collisions that are in the intermediate range between elastic and inelastic, in which there is some
kinetic energy loss but the objects do not stick together.
Problem 5.3 What energy conversion takes place in an inelastic collision
that does not take place in an elastic collision?
Problem 5.4 What are some differences between inelastic and elastic collisions? Give an example of each. Figure 1: A series of snapshots of two different inelastic collisions. Theoretical Background: Conservation of Linear Momentum
Linear momentum, in contrast to energy, is a vector quantity deﬁned as the product of an object’s
mass and its velocity, or
= m .
p
v
(5)
41 Figure 2: A series of snapshots of two different elastic collisions. Therefore, linear momentum has both a magnitude and a direction, and the mathematics of momentum conservation involves breaking momentum vectors into their components, etc. just as we
did with forces.
The Law of Conservation of Linear Momentum is basically just a restatement of Newton’s
Second Law, which states that
p
ext = ∑ ∆ i = ∑ mi i
F
a
i ∆t
i
if the masses remain constant. In the absence of any net external force, ∑
i and therefore ∆ i
p
= 0,
∆t p
∑ i = constant,
i and momentum is conserved. By Newton’s Second Law, therefore, momentum is conserved in any
situation where there is no net external force.
Problem 5.3 Momentum and kinetic energy both depend on mass and velocity and nothing else. So how is it possible that momentum must always
be conserved (in inelastic and elastic collisions, for example) and kinetic
energy is not conserved in inelastic collisions? How is it possible that the
velocities could change in such a way that momentum would be conserved
but kinetic energy would not? Experimental Background: Two Gliders
In the current laboratory, we will be using the glider setup of Laboratory 2, but with one important
difference: there will be two gliders on the track, as shown in Figure 3.
The gliders themselves can be positioned in two ways. If the sides with Velcro on them are facing
42 Figure 3: Air track with two gliders. each other, any collision between the gliders will be an inelastic collision  the gliders will stick
together and travel as one after the collision. However, if the sides with metal bumpers on them
are facing each other, the gliders will bounce off of each other cleanly, and any collision between
them will be an elastic collision.
Problem 5.5 If we collide two gliders on an air track, will the total linear
momentum of the twoglider system be conserved if the environment of the
air track is not perfectly frictionless? Why or why not?
Problem 5.6 If we collide two gliders on an air track, will the total linear
momentum of the twoglider system be conserved if the track is not perfectly level? Why or why not?
The motion of each glider is detected using the same type of ultrasonic sensor that we used in
previous laboratories. Because there are two gliders, there must be two sensors: they are positioned
at opposite ends of the air track and each “sees” the glider closest to it. Experiments
Experiment 1: Inelastic Collisions
We begin by investigating whether kinetic energy and momentum are conserved in the context of
inelastic collisions. We will generate experimental data for two separate inelastic collisions.
1. Turn on the air compressor and place a single glider on the track.
2. Ensure that your track is level by adjusting it so that the glider has no systematic tendency
to move toward one end.
43 Figure 4: Plot of position vs. time for two inelasticallycolliding gliders. 3. Once the track is level, place the second glider on the track. Ensure that the gliders are
situated with the Velcro ends facing each other. Remove any additional brass weights from
the gliders.
4. Open LoggerPro and set it up to collect 5 seconds of data from both sensors simultaneously
at a rate of 10 samples per second by using the template ﬁle impact momentum energy.cmbl.
Practice collecting data and moving the gliders around so you can see which glider corresponds to which sensor. If your data look strange, adjust the tilt angles of the sensors to
ensure they are “seeing” the gliders and not the track or surrounding objects.
5. Weigh both gliders and record their masses in your lab report.
6. Position one glider in the middle of the track, and the other at one end. Begin taking data,
and give the glider at the end of the track a gentle push toward the one in the middle. The
gliders will collide, and your data will look similar to that in Figure 4.
7. Fit lines to the regions of the graph immediately before the gliders collided and after they
collided (see Appendix A Section 1.2). Copy the graph into your lab report, and record the
values of the slopes of these lines in the data table. (What do these slopes represent?)
8. Calculate the total momentum of the system before and after the collision. See Appendix A
Section 2.1 if you need a refresher on how to input formulas.
9. Calculate the total kinetic energy of the system before and after the collision. 44 10. Calculate the ratio of kinetic energy after the collision to kinetic energy before the collision,
and do the same for total momentum. If a particular quantity is conserved, the ratio should
be close to one. (Do you see why?)
11. Repeat this procedure, but add 100 g of brass weights to the glider that is at rest before the
collision (see Figure 5). Figure 5: Attach two 50 g brass weights to the glider’s base; one on either side. Experiment 2: Elastic Collisions
We next investigate whether kinetic energy and momentum are conserved in the context of elastic
collisions. The procedure for Experiment 2 is identical to the procedure for Experiment 1, with
one very important difference. In Experiment 1, you oriented the gliders so that the Velcro ends
were facing each other; this produced the right conditions for inelastic collisions. In Experiment
2, however, you will orient the gliders such that the ends with the metal bumpers are facing each
other. This will produce the right conditions for elastic collisions, in which the gliders bounce off
each other when they collide.
Repeat the procedure for Experiment 1 with the metal bumpers facing each other. Copy both
graphs (with ﬁt lines included) into your lab report and record the glider velocities, masses, and
total momentum and energy before and after each collision.
Experiment 3: Model Fitting and Analysis
Once Experiments 1 and 2 are complete, use the online data collection tool to record the ratios of
kinetic energy and momentum from after the collision to before the collision
ratio = quantity after
same quantity before
45 Figure 6: Plot of position vs. time for two elasticallycolliding gliders. for the inelastic collisions and the elastic collisions. If a given quantity is conserved, the ratio will
be one. However, keep in mind that these are experimental quantities, and therefore are subject to
systematical and statistical error.
1. Enter your data in the online data collection tool as described in appendix A Section 3. We
will use both Charts 1 and 2. Enter the momentum ratios and the kinetic energy ratios into
chart 1 and 2, respectively. The columns of each chart will correspond to the experimental
trial number, e.g. column 1 of chart 1 should contain the momentum ratio for the inelastic
collision with no added mass to the stationary glider.
2. Once all of the class data has been collected, copy the ratios for kinetic energy and total
momentum (after/before) into the data table on your lab report.
3. Calculate a 95% conﬁdence interval for the mean of the kinetic energy and momentum ratios
for all four collision types. (See Appendix A Section 2.2 for useful formulas.)
Problem 5.7 From the class’ experimental results, does it appear that
linear momentum is conserved in trials 1 and 2? How about in trials 3 and
4? Why or why not?
Problem 5.8 From the class’ experimental results, does it appear that
kinetic energy is conserved in trials 1 and 2? How about in trials 3 and 4?
Why or why not? 46 Problem 5.9 Would the following sources of error affect your results
for this experiment (i.e. whether linear momentum/kinetic energy is conserved)? Write a onesentence explanation of why/why not for each.
(a) The gliders experience air resistance.
(b) The track is not level.
(c) The glider that is supposed to be still at the beginning of each trial is
actually moving a bit. 47 ...
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This note was uploaded on 09/13/2011 for the course ECONOMICS 101 taught by Professor Gerson during the Spring '11 term at University of Michigan.
 Spring '11
 Gerson

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