Name:_________________________________ Lab Section: _________ Group Number: __________
Lab Partners: _______________________________________________ Grade:____________
Physics 201
Experiment #7
Collisions in Two-dimensions and Center of Mass
Introduction In this lab you will show that momentum is conserved in a two-dimensional collision. You will need
to take into account the vector nature of momentum. Though vectors may seem to complicate matters, they are
really a convenient tool to simplify a problem by doing good bookkeeping for you. Without vectors, life would just
have no direction! Conservation of momentum in vector-form for a two-body collision can be written as
p1before  p2before  p1after  p2after ,
(1)
u1m1  u 2 m2  v1m1  v 2 m2 .
(2)
which can be expanded to give
Here u is used for initial velocities and v is used for final velocities. We can also think of a multi-body system as
equivalent to one object by invoking the center of mass, see Fig 1. This often significantly reduces the complexity
of a multi-body problem
Figure 1. Example of the center of mass for three different masses
Analytically this can be written as
Rcm 
R1m1  R2 m2  ...RN mN
,
m1  m2  ...mN
(3)
where RN is the position vector of the Nth mass from the origin. The corresponding velocity of the center of mass is
simply the change in position as a function of time
vcm  Rcm 
dRcm
.
dt
(4)
In this lab you will verify conservation of momentum in two ways: by treating the two masses individually and as
one combined mass at the center of mass.
Pre-lab Questions (turn in at start of lab)
1) Does the center-of-mass depend on your choice of origin? Basically, would the physical location of CM in Fig. 1 be
the same if you used a different origin? Could the origin instead have been placed on one of the masses in Fig. 1?
2) How will you turn dots on a piece of paper (the data you take for this lab, showing the paths of colliding particles)
   
into four different velocity vectors u1 , u2 , v1 , v2 . Briefly describe your procedure step-by-step and what tools you
will use. For example, will you measure x- and y- components, and how?
3) What is the advantage to using the center-of-mass frame for this experiment?
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Name:_________________________________ Lab Section: _________ Group Number: __________
Lab Partners: _______________________________________________ Grade:____________
Physics 201
Lab 7
Collisions in two Dimensions and Center of Mass
Introduction In this lab you will show that momentum is conserved in a two-dimensional collision. You will need
to take into account the vector nature of momentum. Though vectors may seem to complicate matters, they are
really a convenient tool to simplify a problem by doing good bookkeeping for you. Without vectors, life would just
have no direction! Conservation of momentum in vector-form for a two-body collision can be written as
p1before  p2before  p1after  p2after ,
(1)
u1m1  u 2 m2  v1m1  v 2 m2 .
(2)
which can be expanded to give
We can also think of a multi-body system as equivalent to one object by invoking the center of mass, see Fig 1. This
often significantly reduces the complexity of a multi-body problem
Figure 1. Example of the center of mass for three different masses
Analytically this can be written as
Rcm 
R1m1  R2 m2  ...RN mN
,
m1  m2  ...mN
(3)
where RN is the position vector of the Nth mass from the origin. The corresponding velocity of the center of mass is
simply the change in position as a function of time
vcm  Rcm 
dRcm
.
dt
(4)
In this lab you will verify conservation of momentum in two ways: by treating the two masses individually and as
one combined mass at the center of mass.
3
Procedure
You will be using a spark-timer similar to the one you used for calculating the motion of a falling body. This time
you will be interested in the motion of objects in a horizontal plane. An air table will provide a “frictionless”
surface. Be sure to level it before you begin your experiments. Recall that the spark-timer fires at a constant rate. Set
the spark timer to 20 Hz. Using carbon paper, you will be able to track the displacement of two metal pucks as a
function of time. To collide them, simply push the pucks into each other with your hands. Start the pucks in adjacent
corners and aim for a collision in the middle of the table. See Fig. 2 for a synopsis of the setup. Try to avoid
touching the table while the spark-timer is sparking, otherwise you’ll get a little jolt.
Figure 2. Two-body collision using the air table and metal pucks
Two-dimensional Conservation of Momentum: Go ahead and collide the two masses. Make sure to avoid any
rotational motion of the pucks. Shoot for at least 10 usable points before the collision and 10 points after. Connect a
line between point A and B in the diagram above and call this the x-axis. Use the right angle tool to draw a y-axis.
1) Why should you avoid rotational motion or does it really matter? Will it change the results of your
calculations for conservation of linear momentum?
Calculate the velocities of both pucks before and after the collisions. Be sure to calculate the associated error with
each.
2) What is the best way to get an accurate measurement of velocities? Taking an average of many points? Plot
a linear fit of some kind? Just using the data points before and after the collision? Something else? Describe
your method here and write down all four velocity vectors, error included.
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Pick one of the final velocities to be “unknown” and use the three other measured values of velocity and values of
the two masses to calculate what it should be. As always, be sure to propagate error.
3) Does your calculated value of the fourth velocity match its measured value? Does the measured value lie
within the error bars of the calculated?
4) Calculate the total kinetic energy before the collision and after. Should it be conserved? If not, what does
this mean about the collision? If not conserved, should it be greater before or after the collision? If you
think it is conserved, quote a percent difference. Use your data to prove your claims.
Analysis using the Center of mass: Your goal is to verify conservation of momentum in the center-of-mass frame.
Calculate Rcm for each set of data points (mass A and mass B for that particular time) and label it on the same paper
with your data. There are no wrong choices of origin, but some make calculations easier than others. Now
calculate vcm . Be careful with your vectors. For your data, the value of the velocity between the Nth point and the
previous is vcmN 
RcmN  RcmN 1
.
t
1) Calculate vcm before and after the collision. Show that momentum is conserved.
2) How does vcm change throughout the entire run? Is this unique to your data or will this be the case for all
collisions using the center-of-mass frame?
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3) In hindsight, would there have been a better place to pick the origin and x- and y- axes to make
measurements and calculations easier? Comment on any symmetry in your data. What do you notice about
where the center of mass points fell for each point in time?
Post-lab Questions:
1) What was the objective of this lab? Do you feel the objective was appropriately achieved? Why or why not?
2) Name the two most significant sources of error in this experiment (Be specific – do NOT say, for example,
“human error” or “equipment limitations”). Are these errors likely to be random or systematic? Explain.
3) Describe some ways that the error in this lab could be reduced, including the possible purchase of additional
equipment.
4) What were the results of this experiment? Was there a physical law you were trying to verify, if so, state that
law. Were you trying to determine the value of some quantity or quantities? If so, what was (were) the value(s)?
Include an estimate of the error in the value(s), written in the form x+x.
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