PHY191 Experiment 5: Elastic and Inelastic Collisions
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Experiment 5
Elastic and Inelastic Collisions
Reading: Bauer&Westfall: Chapter 7 (and 8, for center of mass ideas) as needed
1. Goals
1. Study momentum and energy conservation in inelastic and elastic collisions
2. Understand use of Excel in analyzing data
3. Carry out uncertainty calculations of moderate complexity
2. Theoretical Introduction
The following experiment explores the conservation of momentum and energy in a closed
physical system (ideally: no interaction of measured objects with rest of universe). As you
probably know from the accompanying theoretical course, the conservation of energy and
momentum play an important role in physics and their conservation is a consequence of
fundamental symmetries of nature.
2.1 Momentum
For a single particle (or a very small physical object), momentum is defined as the product of the
mass of the particle and its velocity:
p
mv
(1)
Momentum is a vector quantity, making its direction a necessary part of the data. To define the
momentum in our three-dimensional space completely, one needs to specify its three components
in x, y and z direction. The momentum of a system of more than one particle is the vector sum of
the individual momenta:
p p1 p 2
m1v1 m2 v 2
(2)
The 2nd Newton’s law of mechanics can be written in a form which states that the rate of the
change of the system’s momentum with time is equal to the sum of the external forces acting on
this system:
dp
(3)
F
dt
From here we can immediately see that when the system is closed (which means that the net
external force acting on the system is zero), the total momentum of the system is conserved
(constant).
2.2 Energy
Another important quantity describing the evolution of the system is its energy. The total energy
of a given system is generally the sum of several different forms of energy. Kinetic energy is the
form associated with motion, and for a single particle:
mv 2
KE
(4)
2
PHY191 Experiment 5: Elastic and Inelastic Collisions
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Here v without the vector symbol stands for the absolute value of the velocity,
v x2
v 2y
v z2
In contrast to momentum, kinetic energy is NOT a vector; for a system of more than one particle
the total kinetic energy is the algebraic sum of the individual kinetic energies of each particle:
KE KE1 KE2
(5)
Another fundamental law of physics is that the total energy of a system is always conserved.
However within a given system, one form of energy may be converted to another (such as
potential energy converted to kinetic in the Pendulum experiment). Therefore, kinetic energy
alone is often not conserved.
2.3 Collisions
An important area of application of the conservation laws is the study of the collisions of various
physical bodies. In many cases, it is hard to assess how exactly the colliding bodies interact with
each other. However, in a closed system, the conservation laws often allow one to obtain the
information about many important properties of the collision without going into the complicated
details of the collision dynamics. In this lab, we will see in practice how the conservation of
momentum and total energy relate various parameters (masses, velocities) of the system
independently of the nature of the interaction between the colliding bodies.
Assume we have two particles with masses m1 , m2 and speeds v1i and v2i which collide, without
any external force, resulting in speeds of v1 f and v 2 f after the collision ( i and f stand for initial
and final). Conservation of momentum then states that the total momentum before the collision
Pi is equal to the total momentum after the collision Pf :
Pi
m1v1i
m2 v 2i ,
Pf
m1v1 f
m2 v 2 f
and
Pi
Pf
(6)
2.4 Elastic and inelastic collisions
There are two basic kinds of collisions, elastic and inelastic:
2.4.1 In an elastic collision, two or more bodies come together, collide, and then move apart
again with no loss in total kinetic energy. An example would be two identical "superballs",
colliding and then rebounding off each other with the same speeds they had before the collision.
Given the above example conservation of kinetic energy then implies:
1
1
1
1
(7)
m1v12i
m2 v 22i
m1v12f
m2 v 22 f or KEi KE f
2
2
2
2
2.4.2 In an inelastic collision, the bodies collide and (possibly) come apart again, but now some
kinetic energy is lost (converted to some other form of energy). An example would be the
collision between a baseball and a bat. If the bodies collide and stick together, the collision is
called completely inelastic. In this case, all of the kinetic energy relative to the center of mass of
the whole system is lost in the collision (converted to other forms).
PHY191 Experiment 5: Elastic and Inelastic Collisions
a)
b)
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In this experiment you will be dealing with
a completely inelastic collision in which all kinetic energy relative to the center of mass
of the system is lost, but momentum is still conserved, and
a nearly elastic collision in which both momentum and kinetic energy are conserved to
within a few percent.
2.5 Conservation laws for macroscopic bodies
So far we were talking about the system of point-like particles. However, the conservation of the
momentum is also valid for macroscopic objects. This is because the motion of any macroscopic
object can be decomposed into the motion of its center of mass (which is a point in space) with a
given linear momentum, and a rotation of the object around this center of mass. Then, the
conservation of the linear momentum is again valid for the motion of ideal point masses located
at the center of the mass of each of the objects. However, some of the linear kinetic energy can
be transformed into the rotational energy of the objects, which should be accounted for in a real
experiment.
2.6 Kinetic Energy in Inelastic Collisions.
It is possible to calculate the percentage of the kinetic energy lost in a completely inelastic
collision; you will find that this percentage depends only on the masses of the carts used in the
collision, if one of the carts starts from rest.
After the completely inelastic collision, the carts move together, so that
v1 f v 2 f v3
The initial KE is given by:
m1v12i m1v 22i
KEi
. But, since v 2 i 0
2
2
m1v12i
KEi
2
The final KE is given by:
m2
2 m
KE f v 3 1
2
From conservation of momentum:
m1v1i m2 v 2i (m1 m2 )v3
m1v1i
(m1
or, since v 2i
m 2 )v 3
(8)
(9)
0
(10)
Since the collision is inelastic, the initial KE is not equal to the final KE. You could use
( KE f KEi )
equations (8), (9), and (10) to obtain an expression for DK (%)
. Hint: define x =
KE i
m1/(m1+m2) and use it to eliminate v3.
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3. Experimental setup
We will study the momentum and energy conservation in the following simplified situation:
a)
we will look on the collision of only 2 objects;
b)
the motion of these objects will be linear and one-dimensional, so that we can choose the
reference frame in such a way that only x-components of the objects’ momenta are nonzero; the sign of these components depends on the direction of the motion;
c)
the experimental apparatus can be set up in a way to almost completely eliminate the net
external force on the system.
Our objects will be two carts of different masses, with one initially at rest. The carts move on an
air track, which ensures that the motion is one-dimensional and reduces the friction between the
carts and the surface. The velocities of the carts can be measured with the help of the photogates,
which are described in more details below.
Before the beginning of the measurements, spend at least 15 minutes to figure out which external
factors can disturb the motion of the carts on the track, and what you should do to reduce or
eliminate these factors. Remember, the successful completion of this lab strongly depends on
your ability to create an almost closed system. Make a few practice trials to see if you can
achieve an unperturbed one-dimensional collision of the carts. Adjust the level of the air track
and the power of the air supply if necessary.
Questions for preliminary discussion
3.1
Draw a diagram of all forces acting on each cart when they collide. Which forces will
influence the total P and KE most?
3.2
In our experiment, can we achieve a completely elastic collision? a completely inelastic
collision?
3.3
In an inelastic collision in a closed system, can some of the total momentum be lost?
Some of the kinetic energy?
3.4
In an elastic collision in a closed system, can some of the total momentum be lost? Some
of the kinetic energy be lost?
3.5
If kinetic energy is lost, where does it go? Does conservation of energy apply?
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Measurements
4. Inelastic collisions
4.1
Techniques. In the first part of the lab we make sure that after the collision the carts
stick together and move with some velocity common to both masses. Thus, we have to measure
the velocity of cart 1 before the collision and the common velocity of the carts 1 and 2 after the
collision. For this purpose, we use two photogates (see Figure 1). Each of them allows measuring
the time it takes the cart to go through it. The speed is calculated by dividing the length of the fin
on the cart 1 by the measured time; but to turn speed into velocity, as required, you have to pick
a + direction.
Figure 1: Initial state of the carts before inelastic collision (fin on cart 2 can be removed).
Warning: your carts should be balanced, unlike the ones in this figure!
Position cart 2 close to the gate 2 and set the photogate timer to "GATE" mode and the memory
switch in “ON” position. In this mode the photogate will display the first time interval measured.
Subsequent measurements will not be displayed (only the first one is), but the times are added in
the memory. By pushing the “READ” switch you can display the memory contents, which is the
sum of all measurements. Example: the initial reading for cart 1 (the time that it took to pass
through the gate 1) is 0.300 seconds. Cart 1 collides with cart2 and they go together through the
photogate 2 (Figure 2). Suppose it now takes 0.513 seconds. The display will remain at 0.300,
but the memory will contain .300 + 0.513=0.813 seconds. To find the second time, you have to
subtract the first time from the contents of the memory. Try this out by moving the cart through
the gate by hand a few times.
4.2 Uncertainty Estimation: First we test to see whether some of our assumptions are correct
(the two timers give the same answer, the track is level, friction is negligible, direction doesn’t
matter). Perform some trials with a single cart in which no collision occurs. Do this at different
speeds, and in different directions. Record the results in your notebook in appropriate tables.
Did you conserve momentum? Was there a bias? What time uncertainty would you deduce
from these measurements? What conclusions do you draw?
4.3 Prediction: Which case below will the change the final Kinetic Energy most? In your
report comment on whether your predictions were correct, and if not, why.
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4.4 Data Do 3 sets of inelastic collisions consisting of 2 trials each. Vary the masses of the carts
by adding the masses (small metal disks) to them. In these measurements, use the needle and
putty bumpers and measure the initial and final velocities for the following sets of masses:
Trial 1+2: no mass disks on cart 1, 4 mass disks on cart 2;
Trial 3+4: 2 mass disks on cart 1, 2 mass disks on cart 2;
Trail 5+6: 2 mass disks on cart1, no mass disks on cart2.
In each measurement, you need to find all the initial and final masses and velocities, and use
them to calculate the initial and final total momentum and kinetic energy. Make tables in your
lab book to organize your recordings. You may neglect the uncertainty of all masses.
In R:exp6, you will find 2 preset datasheets, Inelastic and Elastic. Open them and save into your
section’s folder. The spreadsheet has some entries for fractional uncertainties ( Dp(%), for
example). These should be displayed in % either by multiplying the relevant fraction by 100, or,
preferably, by using the % formatting button in Excel. When you have completed the
spreadsheets, print them showing the numbers, then save them. There are many uncertainty
calculations. You should explain in your lab book or report how you calculated them. An
alternative is to re-open your spreadsheet, save it with a new name, use the ctrl-` key to display
formulas (handy while checking formulas) and print again, showing the formulas used. It won’t
be possible to read the formulas unless you adjust the column width, and use page setup to print
landscape, to fit to 2 pages wide x 1 tall. You can use Print Preview before printing to check.
4.4 Hints The correct calculation of the uncertainty of D(%) is complicated because the initial
value for p or K is in both the numerator and denominator. However, for our purposes, it is
sufficiently close to use Dp(%)
D/Pi and a similar formula for K.
(Optional) If you wish you could add auxiliary cells to help you with uncertainty
calculations. For example, you could also calculate v/v or P/P.
(Sanity checks for uncertainties) Check whether your uncertainty calculations are making
sense! For independent and random uncertainties, you expect increasing values of absolute and
fractional uncertainties as your calculation proceeds (unless your calculation involves fractional
powers < 1). This is discussed in the Reference Guide section on Uncertainty Calculations.
5. Elastic collisions
5.1 Techniques. In an almost elastic collision, the main difference from the previous part of the
lab is that after the collision the carts move separately. The rubber band bumpers allow carts to
collide with almost no conversion of the kinetic energy into the other forms of energy.
As before, cart 2 initially stays at rest, and before the collision we have to measure only the
velocity of the cart 1 v1i (Figure 3). However, after the collision we have to measure the
velocities of both carts, v1 f and v 2 f (Figure 4). Thus, all in all we have to measure three times (
t1i , t1 f , t 2 f ), while the photogate system can simultaneously measure only two of them.
We can get out of this situation if, after the measurement of the initial time t1i , but before the
collision, we reset the timer. You have to make several practice trials to quickly remember and
reset the contents of the timer before the carts collide. Then, we can again see the contents of the
timer display and the memory to find t1 f and t 2 f .
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Figure 3: The initial state of the carts before elastic collision.
Figure 4: The final state of the carts after elastic collision.
5.2 Prediction: Write in your lab book for each case mentioned in section 5.3 below the
predicted direction and speed after the collision of the initial-moving cart:
Forward or backward?
Faster or slower than the cart initially at rest?
In your report comment on whether your predictions were correct, and if not, why.
5.3 Data Measure the fin length on the two carts. The experiment will be done with cart 2
initially at rest. You will do 6 trials with the following choices of m1 and m2 :
Trial 1+2: no mass disks on cart 1, 4 mass disks on cart 2
Trial 3+4: 2 mass disk on cart 1, 2 mass disks on cart 2
Trail 5+6: 4 mass disks on cart 1 no mass disk on cart 2
5.4 Hints. Pay attention to the sign of the velocities, which depends on the direction of motion
of the cart. If one of the carts goes backward with respect to your chosen + direction, how will
you make sure the velocity is calculated as negative?
If the percentage change in momentum or kinetic energy before and after the collision is greater
than 10%, repeat the measurement more carefully (collide slower/faster, etc.). Since the
datasheet is set up it is easy to see whether momentum/energy is better conserved with every trial
you do.
If one of the times you measure is too long for the timer to measure, substitute a large number for
the time in your spreadsheet.
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6 Questions to be discussed
6.1
Create a graph (one for inelastic and one for elastic collisions) of the relative change of
the total momentum D p (%) ( Pf Pi ) / Pi versus the number (1-6) of the measurement.
6.2
6.3
6.4
6.5
This is just to show graphically the various values of Dp(%). Show the uncertainty of
Dp(%) with the help of error bars. Also, show the theoretical prediction for Dp(%) .
According to the graphs, was the total momentum conserved in the collisions? Use the
“two standard deviations” rule to justify your answers. Was there any difference in how
well momentum was conserved in elastic vs. inelastic collisions? If the momentum was
not conserved, discuss the reason why. What did you do to try to improve the momentum
conservation results?
Did you correctly predict the motion of the carts in the elastic collision? If not, why?
For the inelastic collision, plot the kinetic energy fractional change, DK(%) vs.
m1
x
. Show the error bars for DK(%). What are the slope and intercept of a
m1 m2
straight line fit to these data? Does a straight line fit these data reasonably? Did you
correctly predict the trial with the largest fractional change? Why or why not?
Make and analyze a DK(%) graph for the nearly elastic collision (as you analyzed Dp(%)
above in steps 6.1-6.3). What was the average loss of the kinetic energy in this part of the
experiment? Did you achieve the goal of 10% loss of KE in the nearly elastic collisions?
Which potential systematic errors or setup problems is the loss most sensitive to?
7. Systematic Errors
Professional scientists deal with systematic errors in successive levels of sophistication.
Level 1:Think of possible errors. This is one of the hardest parts! That’s because doing so
means seeing where our assumptions break down. The best way is to go through the
measurement step by step and identify where things could have been otherwise than we wished.
The result is a laundry list of possible systematic effects. Some examples of our assumptions:
Setup: We assumed that the track was straight and level. Carts were balanced.
Timing: The velocities were constant while the carts moved through the gate and until the
collision.
Collision: The system of the 2 carts is closed: no momentum (nor energy) enters or leaves the
system consisting of the two carts during the collision.
Question 7.1 List ways in which these assumptions might have been violated.
Level 2: Sign of effect. For each violation of our assumptions, try to understand the sign of its
effect on our final result. For example, if the carts weren’t balanced, then the air track would
tend to add momentum of a particular sign to the cart. It often helps to exaggerate (mentally, or
by measuring) the effect to see which way it should affect your results.
7.2 What sign would each of your effects have on momentum or energy conservation? Or
could it be of either sign?
Level 3: Estimating the size of the effect. Sometimes we can perform a side-experiment and
actually measure the effect and correct for it. More commonly, we can only provide a bound
“well it should really have been at worst this big”. For example, leveling errors didn’t give the
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carts enough extra momentum to go the length of the track in 10 seconds. With the mass, that
bounds the change of momentum; that over the smallest initial momentum bounds the % error.
7.3 How big an effect could the air track effect be on your momentum conservation data?
7.4 How big an effect would th ...

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