PHY 131/221
Conservation of Momentum in Two Dimensions
(revised 12/21/07 J)
Name: ________________________
Date: ______________
Lab Partners: __________________________________________
Introduction: Momentum, mass times velocity, is extremely important when analyzing the motion of a
system of objects, especially when considering the results of a collision. As you learned in class, momentum
is a vector, every object of a system has momentum (although it might be zero) and the momentum vectors
of all of the objects in a system can be added together to give something we call the total momentum of the
system. (This is similar to having several forces acting on an object and adding all these forces to get the
total, or net, force acting on the object.)
Why is momentum so important? Because, as you may have learned in class, the total momentum vector of a
system is constant as long as no external forces are acting on the system or as long as the total external force
acting on the system is zero. This is the law of Conservation of Momentum. This does not mean that each
object in the system will have a constant momentum, but that the total of all momenta added together will be
constant. Individual momenta may change, but the total will remain the same as time goes on.
A system might actually consist of a single object. If there are no forces acting on the object then the object’s
momentum will be constant. One source of confusion could be that even though the object might be moving,
its momentum is still constant. Remember that momentum is mass times velocity so if velocity is constant so
too is momentum. One way to see this is to draw a dot to represent the location of the object at various points
in time and at each dot draw an arrow to represent the object’s velocity, or momentum, with the tail of the
vector on the dot (see figure 1). Even though you have drawn several dots with an arrow on each, all these
arrows (vectors) are equal because their lengths are all equal and also their directions are equal. So you do
not really need to draw an object’s momentum vector with its tail at the location of the object. All you really
need is a single vector drawn somewhere on your paper that represents the direction and magnitude of the
momentum at all time. Only when the momentum changes should you draw separate momentum vectors at
different locations on your paper.
Figure 1
In this lab you will analyze several collisions that take place in two dimensions. The momentum vectors will
therefore have up to two components. In all of these collisions one ball will be moving and the other will be
stationary before the collision. How do we determine the velocity, or momentum, of the moving ball before
the collision? We will do this by letting the ball roll down a ramp as shown in Figure 2. When the ball gets to
the bottom of the ramp, just before colliding with the stationary ball, it will have a speed that can easily be
determined as outlined in Part I.
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Figure 2
Part I: Initial Momentum of Moving Ball and Initial Momentum of the System. Clamp
your ramp to the edge of a table so that the end of the ramp with the adjustable ball holder is hanging off the
edge of the table. Tape a large piece of paper (approximately 3 ft by 3 ft) onto the floor so that one edge of
the paper is directly below the edge of the table.
Drop a plumb line down from the end of the ramp, centered on the ramp, so that the plumb bob touches the
paper. Mark this spot on your paper and label it with a number 1. This point will help define the initial
direction of travel of the moving ball and it will indicate the horizontal position of the moving ball just
before the collision. Your instructor will demonstrate this. You will use carbon paper to determine where the
balls land after the collision. You might ask why we are letting the balls fall when we are really trying to
study a two dimensional collision. Letting the balls fall introduces a third dimension. Shouldn’t we just let
the balls roll on a flat horizontal surface so that the motion is entirely in the horizontal plane? We could do
this if we had a perfectly horizontal surface and rolling friction was negligible, but we would still need to
figure out a way to measure the balls’ speeds. Our solution is to use the acceleration due to gravity along
with the vertical and horizontal distances traveled to determine the balls’ velocities immediately after the
collision. This is just like the projectile motion lab where we fired a ball horizontally to determine its launch
speed. We will thus avoid any error due to a horizontal surface not being perfectly horizontal, errors due to
rolling friction, and difficulty in measuring speeds directly.
The velocity of the moving ball before the collision multiplied by the ball’s mass will give the ball’s
momentum before the collision. Since the other ball is stationary the moving ball’s momentum is equal to the
system’s total momentum and this total momentum will equal the sum of both balls’ momenta just after the
collision.
To measure the moving ball’s initial speed we will let the ball roll off the ramp without colliding with the
other ball. As soon as the ball leaves the ramp it will start falling under the force of gravity. The distance
from the ball’s landing spot to the spot directly below the ramp that you marked previously will equal the
ball’s horizontal distance traveled during free fall.
Measure the mass of the steel balls and enter this in the data section.
Allow one ball to roll down the ramp several times, starting from the same initial height each time, and use
carbon paper to mark the landing position. Determine a point on the paper that represents the average of
these landing spots and label this point with a number 2. Draw a line from this average landing spot, point
2, to the spot under the edge of the ramp, point 1. This line represents the direction of the moving ball’s
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momentum or velocity. Measure the length of this line and record it in the data section. This line will also be
our x-axis. Our y-axis will then be perpendicular to this line.
To determine the ball’s horizontal speed we need to know how much time it took for the ball to travel this
distance. This time is equal to the time it took for the ball to fall the distance it fell. Measure the vertical fall
distance and record in the data section. (From where do you measure the initial height of the ball?) Calculate
the time it took for the ball to fall and enter this in the data section. Finally, calculate the horizontal speed of
the ball and enter this in the data section. Be sure to show your calculations and do not round any calculation
until you have calculated the horizontal speed. (Keep at least four decimal places in all your calculations and
remember to include as many significant figures as possible in all of your measurements.)
Finally, calculate the ball’s horizontal momentum and, for PHY 221 students, express it in unit vector
notation in the data section. Remember that this momentum is the total momentum of the system before and
after the collisions that we will now consider.
Part II: Collision With Identical Ball at about 20 degrees.
Now adjust the stationary ball holder so that the moving ball collides with the stationary ball at a slight angle
of about 20 degrees as shown in figure 3. This angle is not critical we just don’t want a head-on collision, nor
do we want a glancing collision. Also, adjust the height of the ball holder so the balls are at the same
vertical position when they collide. Be sure to adjust the stationary ball holder so that the center of the
moving ball is right at the edge of the ramp when it just touches the stationary ball. Doing this will
ensure that the moving ball is directly over point 1, on your paper, when the collision occurs. Tighten the ball
holder in place. See Figure 3 and 4.
Figure 3
Figure 4
Drop a plumb line directly below the ball holder to make a mark on the paper directly below the initial
position of the stationary ball. Label this mark, “S1” (S for stationary).
Now place the stationary ball on the ball holder and hold the moving ball at the top of the ramp. Do a trial
collision to see where the balls will land. Place pieces of carbon paper at the landing locations and do three
trials to create three landing marks for each ball. Find an average for these locations and mark these points
“S2” for the initially stationary ball, and “M2” for the initially moving ball.
Draw a line between S1 and S2, and between point 1 and M2 as in figure 5. These lines give the distance
traveled horizontally after the collision. Measure the length of these lines and enter in the data section.
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Figure 5
To determine the speeds of the balls just after the collision we simply divide the above distances by the time
of fall for each ball. Does each ball take the same time to fall? Are these times the same as the time
calculated in Part I? Calculate the speeds of each ball immediately after the collision and enter these speeds
in the data section. Also, calculate the momenta of the balls just after the collision.
We will do further analysis of this data later.
Part III: Collision With Identical Ball at a glancing angle.
Now adjust the stationary ball holder so that the moving ball collides with the stationary ball at a glancing
angle. In other words, the moving ball should just slightly make contact with the stationary ball. Don’t make
the collision too “glancing” otherwise the stationary ball may fall and hit the ball holder on its way down.
Also, adjust the height of the ball holder so the balls are at the same vertical position when they collide, and
be sure the center of the moving ball is directly over the edge of the ramp when it makes contact with the
stationary ball. Tighten the ball holder in place.
Drop a plumb line directly below the ball holder to make a mark on the paper directly below the initial
position of the stationary ball. Label this mark, “S3”.
Now place the stationary ball on the ball holder and hold the moving ball at the top of the ramp. Do a trial
collision to see where the balls will land. Place pieces of carbon paper at the landing locations and do three
trials to create three landing marks for each ball. Find an average for these locations and mark these points
“S4” for the initially stationary ball, and “M4” for the initially moving ball.
Draw a line between S3 and S4, and between point 1 and M4. These lines give the distance traveled
horizontally after the collision. Measure the length of these lines and enter in the data section.
To determine the speeds of the balls just after the collision we simply divide the above distances by the time
of fall for each ball. Does each ball take the same time to fall? Are these times the same as the time
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calculated in Part I? Calculate the speeds of each ball immediately after the collision and enter these speeds
in the data section. Also, calculate the momenta of the balls just after the collision.
We will do further analysis of this data later.
Part IV: Analysis of Data From Part II.
To verify that momentum is indeed conserved we can use either an algebraic method, or a graphical method.
We will do both! 
Algebraic Method: This method involves breaking vectors into their components and showing that the xcomponents and y-components are separately conserved. In other words, the initial momentum of the system
has only an x-component (you calculated this in Part I). The x-components of the two balls after the collision
should add up to equal the initial x-component, and the y-components should add up to equal zero (i.e. the ycomponents should be equal in magnitude, but opposite in direction).
The lines S1-S2 and 1-M2 were measured as distances, but these lines can also be thought of as the
momentum vectors of the two balls just after the collision. This is true because we determined momentum by
dividing these distances by time and then multiplying by mass. Since the time and mass are equal for both
balls then the momenta are different only because the distances are different. So the distances give us a direct
indication of the momenta. Read the last three sentences again, and again, to be sure you understand
what is being explained.
Measure the angle of line S1-S2 relative to the x-axis. Measure the angle of line 1-M2 relative to the x-axis.
Calculate the x- and y-components of these momenta (remember to include a negative sign where
appropriate based on your choice of the positive y-direction). Enter this data in the data section.
Also, in the data section write down the x- and y-components of the moving ball’s momentum before the
collision. Remember that this momentum is equal to the total system momentum before the collision and
should also equal the total system momentum immediately after the collision.
From your measurements and calculations is momentum conserved in this collision? Explain your answer in
the data section.
Graphical Method: On your paper draw an arrow head at point S2 and M2 and point 2. These arrow heads
represent the heads of the momenta vectors. Construct a triangle showing addition of the S-momentum
vector and the M-momentum vector. Measure the length and direction of the resultant vector (the total
momentum after the collision) and compare this to the length and direction of the initial momentum vector
before the collision. Is momentum conserved in this collision. Explain your reasoning in the data section.
Finally, measure the angle between the momentum vectors of the two balls immediately after the collision. Is
this angle what you expected it to be? Explain in the data section.
Part V: Analysis of Data From Part III.
Algebraic Method:
Measure the angle of line S3-S4 relative to the x-axis. Measure the angle of line 1-M4 relative to the x-axis.
Calculate the x- and y-components of these momenta (remember to include a negative sign where
appropriate based on your choice of the positive y-direction). Enter this data in the data section.
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Also, in the data section write down the x- and y-components of the moving ball’s momentum before the
collision. Remember that this momentum is equal to the total system momentum before the collision and
should also equal the total system momentum immediately after the collision.
From your measurements and calculations is momentum conserved in this collision? Explain your answer in
the data section.
Graphical Method: On your paper draw an arrow head at point S4 and M4 and point 2. These arrow heads
represent the heads of the momenta vectors. Construct a triangle showing addition of the S-momentum
vector and the M-momentum vector. Measure the length and direction of the resultant vector (the total
momentum after the collision) and compare this to the length and direction of the initial momentum vector
before the collision. Is momentum conserved in this collision. Explain your reasoning in the data section.
Finally, measure the angle between the momentum vectors of the two balls immediately after the collision. Is
this angle what you expected it to be? Explain in the data section.
Be sure to put your names on the large sheet of paper and turn it in with your lab write-up.
Proceed to the Questions section.
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Data
Part I: No Collision
Mass of ball = ______________(kg)
Horizontal Distance Traveled of Moving Ball = ________________(m)
Vertical Distance of Fall = _____________(m)
Time to Fall = _________________(s)
Horizontal speed of Moving Ball = _________________ (m/s)
Ball’s Horizontal Momentum = __________________(kg m/s)
Show calculations below:
Part II: Collision With Identical Ball at about 20 degrees.
Distance from S1 to S2 = _______________(m)
Distance from point 1 to M2 = ______________(m)
Horizontal Speed of Stationary Ball Just After Collision, VS = _________________(m/s)
Horizontal Speed of Moving Ball Just After Collision, VM = _________________(m/s)
Magnitude of Momentum of Stationary Ball Just After Collision, PS = ______________(kg m/s)
Magnitude of Momentum of Moving Ball Just After Collision, PM = ________________(kg m/s)
Show calculations below:
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Part III: Collision With Identical Ball at a glancing angle.
Distance from S3 to S4 = _______________(m)
Distance from point 1 to M4 = ______________(m)
Horizontal Speed of Stationary Ball Just After Collision, VS = _________________(m/s)
Horizontal Speed of Moving Ball Just After Collision, VM = _________________(m/s)
Magnitude of Momentum of Stationary Ball Just After Collision, PS = _________ _____(kg m/s)
Magnitude of Momentum of Moving Ball Just After Collision, PM = ________________(kg m/s)
Show calculations below:
Part IV: Analysis For Part II.
Algebraic Method:
Angle for S-momentum vector = ______________(degrees)
Angle for M-momentum vector = _________________(degrees)
x-component of S-momentum = ________________(kg m/s)
x-component of M-momentum = ________________(kg m/s)
y-component of S-momentum = ________________(kg m/s)
y-component of M-momentum = ________________(kg m/s)
Sum of x-components of S- and M-momenta = __________________(kg m/s)
Sum of y-components of S- and M-momenta = ___________________(kg m/s)
x-component of total momentum before collision from part I = ______________(kg m/s)
y-component of total momentum before collision from part I = ______________(kg m/s)
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Is momentum conserved in this collision? Explain and show calculations to justify your answer.
Graphical Method:
Is momentum conserved in this collision based on graphical addition of the momentum vectors? Explain.
What is the angle between the two balls’ momentum vectors just after the collision? _________(degrees)
Is this what you expected? Why or why not.
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Part V: Analysis For Part III.
Algebraic Method:
Angle for S-momentum vector = ______________(degrees)
Angle for M-momentum vector = _________________(degrees)
x-component of S-momentum = ________________(kg m/s)
x-component of M-momentum = ________________(kg m/s)
y-component of S-momentum = ________________(kg m/s)
y-component of M-momentum = ________________(kg m/s)
Sum of x-components of S- and M-momenta = __________________(kg m/s)
Sum of y-components of S- and M-momenta = ___________________(kg m/s)
x-component of total momentum before collision from part I = ______________(kg m/s)
y-component of total momentum before collision from part I = ______________(kg m/s)
Is momentum conserved in this collision? Explain and show calculations to justify your answer.
Graphical Method:
Is momentum conserved in this collision based on graphical addition of the momentum vectors? Explain.
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What is the angle between the two balls’ momentum vectors just after the collision? _________(degrees)
Is this what you expected? Why or why not.
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Questions
1.
Are the collisions studied in this lab elastic, or inelastic? How do you know?
2.
Explain what would happen if you setup the stationary ball so that you had a head-on collision. Where
would the balls land? Explain.
3.
What was the greatest source of error in this experiment? Was this error random, or systematic?
4.
Suppose you did a collision where the stationary ball was at about 20 degrees from the initial
direction of the moving ball, and the stationary ball was half the mass of the moving ball. Would momentum
still be conserved? Would you measure the same angle between the final momenta as you did in the
graphical methods of parts IV and V? Explain.
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