Alabama Science in Motion
Conservation of Momentum in 2-Dimensions
Conservation of Momentum in 2 Dimensions
EQUIPMENT NEEDED
Item
Projectile Launcher with Collision Attachment
Top Load Balance (Optional one per class set)
Steel Balls
Plastic/glass Ball (Optional)
Small C-Clamp or Table Clamp
QTY
1
1
2
1
1
Item
Protractor
Plumb Bob
Carbon Paper
Meter Stick
Butcher Paper or equivalent
QTY
1
1
3
1
1
Question
How do Atomic Physicists determine the mass or the speed of a particle that you can’t
even see with your eyes?
Background
As a model, consider the collision of two objects in the normal three dimensions of
forward and backward (X), left, right (Y), and up, down (Z). In this activity, a ball is fired
horizontally (in the X, Y plane) by the projectile launcher. Just after it clears the end of
the barrel of the launcher, it strikes a glancing blow off the side of another ball initially at
rest and at the same height (Z). After the collision, the two balls continue to move
downward (Z-direction) under the influence of gravity. Since the height is the same for
both objects when the collision occurs, there is no transfer of momentum in the Zdirection. However, the momentum is transferred between the balls in both the X and Y
directions of the horizontal plane. By analyzing the impact data (where the balls
collided and then hit the ground), you can determine their final velocities. With their
masses and velocities, you can demonstrate that momentum is conserved
independently in the X and Y
directions.
Y-Axis
Let the X-axis direction to be
the same as the initial direction
of the velocity of ball #1. The
Y-axis directions will be in the
horizontal plane at 90° to the
X-axis. The momentum before
the collision is
pbefore = m1vo
M2 final
X-Axis
θ2
θ1
where vo is the initial velocity of
m1 in the x-direction only. The
momentum after the collision
has both x and y components:
pafter (x) = m1v1cosθ1 + m2v2cosθ2
pafter (y) = m1v1sinθ1 - m2v2sinθ2
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M1 initial
M1 final
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Alabama Science in Motion
Conservation of Momentum in 2-Dimensions
In the X-direction, the momentum before the collision must equal the momentum in the
x-direction after the collision
m1vo = m1v1cosθ1 + m2v2cosθ2
(1)
Since there is no initial momentum in the y-direction before the collision, conservation
of momentum requires that there is no net momentum in the y-direction after the
collision. Therefore,
0= m1v1sinθ1 + m2v2sinθ2
OR
(2)
m1v1sinθ1 = -(m2v2sinθ2)
Note: the minus sign indicates that the final velocities in the y-direction must be
opposite directions.
PROCEDURE
1. Clamp the Mini Launcher to the table as
demonstrated by the instructor.
2. Adjust the barrel angle to zero to launch
horizontally.
3. Mount the collision accessory onto the front of the
launcher with the side that says “Use This Side for
Mini Launcher” facing upward. Slide the
attachment back along the launcher until the tee is
about 2 cm in front of the muzzle as shown. The
height of the tee positions the two balls at the same
elevation at the time of the collision.
4. Tape a plumb line below the tee that is just long enough to reach the table (or floor if
firing to the floor). This plumb line will allow you to mark on paper where the
collision occurred.
5. Tape a large piece of paper below the launcher so that the plumb line hanging from
the bottom tee can mark the location of the collision. The paper must be large
enough to record the impact of the launched ball with no collision and wide enough
to record the impact of both balls after the collision.
6. Now rotate the collision bracket so that the tee is to the side of the barrel.
To complete the set up, you will test fire the projectile ball without a collision in order to
make sure the ball will land on the paper. You will also need to test fire the projectile
with a collision to make sure that both the projectile (m1) and the target ball (m2) will
land on the paper after the collision. Warn your neighbors before each shot. Make
sure someone is responsible for tracking each ball at all times. Note approximately
where each ball lands so that carbon paper can be placed over the point of impact.
During the experimental shots, the carbon paper will mark the landing point of each ball.
7. Warn your neighbors before firing the projectile without a collision. Load the ball
into the launcher at the short range setting (one click). Station an observer to note
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Alabama Science in Motion
Conservation of Momentum in 2-Dimensions
the spot where the Ball #1 (metal ball) strikes the floor (or table) during a test shot
(No Collision this time). Make sure the projectile ball lands on the paper and that
someone is responsible for where the ball goes after it lands! If the ball does not
land on the paper, consult your instructor.
8. Warn your neighbors before firing the ball with a collision. Place the projectile ball
into the launcher but DO NOT use the plunger to load to the short range yet. Place
the target ball on the tee and rotate the tee so that the edge of the target ball will just
make contact with projectile ball during the collision. Tighten the collision bracket
into position.
SAFETY NOTE: DO NOT LOOK INTO THE LAUCHER to make this adjustment.
Keep your face well above the level of the loaded launcher. Make this adjustment
from beside and above the launcher and without the tension in the launcher spring.
9. When the target ball is positioned for collision, gently remove the target ball without
moving the collision accessory.
10. Use the plunger to load the projectile ball to the short range setting (one click).
Return the target ball to the tee. Warn your neighbors before test firing the projectile
with a collision. Station an observer to note the spots where the each ball strikes
the floor following the test shot with a collision.
11. Fire you launcher for the test run with a collision. Make sure that both balls lands on
the paper and that someone is responsible for where each ball goes after it lands! If
one or both balls do not land on the paper, consult your instructor.
12. Place (DO NOT TAPE) a piece of carbon paper over the three landing spots. Make
sure the marking side of the carbon paper is down.
Recording your experimental trials:
13. Without moving the collision accessory, load the projectile ball to the short range
setting and launch the projectile ball WITHOUT a collision. Repeat for a total of 5
trials. Make sure to warn your neighbors before each shot and keep up with the ball
after it hits the paper! These 5 trials will provide your initial momentum data.
14. Without moving the collision accessory, load the projectile to the short range setting,
set the target ball on the tee and launch the projectile WITH a collision. Repeat for
a total of 5 trials. Make sure to warn your neighbors before each shot and keep up
with the balls after they hit the paper! These 5 trials will provide your final
momentum data.
15. Use the plumb bob to mark a point directly beneath the tee where the collision
occurred. This will be the origin for your measurements and sketch.
16. Measure the change in height for the projectiles and record the value in your data
table.
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Alabama Science in Motion
Conservation of Momentum in 2-Dimensions
Analyzing the data:
1. Look at the impact marks for the initial momentum (NO Collision) trials. Visually
estimate the center of these five impact points. Place an x at this center point.
2. Draw a line connecting the origin below the plumb line to the x marking the center of
the initial momentum data points. This will be your initial momentum line AND your
X-axis.
3. To draw the Y-axis line, set a protractor on the origin created by the plumb bob so
that the 90 mark lies along the X-axis or Initial Momentum line. Extend the 0 and
180 degree lines of the protractor to make your Y-axis in both directions. Make sure
your Y-Axis is exactly 90 degrees (perpendicular) to your X-Axis. You cannot do this
without a protractor and ruler.
4. Now, look at the impact marks created by m1, the projectile ball after the collision.
Visually estimate the center of these five impact points. Place an x at this center
point. Label this point “m1 final momentum.”
5. Draw a line connecting the origin to the point “m1 final momentum.” Label this line
R1. Measure and record the length of R1 on your paper and in your data table.
6. Look at the impact marks created by m2, the target ball after the collision. Visually
estimate the center of these five impact points. Place an x at this center point.
Label this point “m2 final momentum.”
7. Draw a line connecting the origin to the point “m2 final momentum.” Label this line
R2. Measure and record the Length of R2 on your paper and in your data table.
8. Use the protractor to measure the angles (θ1 and θ2). Measure from the X-Axis to
R1 and R2 respectively.
9. Calculate the velocities: You should recall from your study of projectile motion that
the velocity of a projectile fired horizontally can be found using the equation
V=range/time of flight. You measured your ranges above. Recall from your study of
projectile motion that time of flight for a projectile fired horizontally is found using the
equation ∆t = (2∆h/g)1/2. Since ∆h is the same for all projectiles in the activity, they
should all have the same time of flight.
10. Answer the questions following the Data and Calculations summary tables.
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Student Name:
Conservation of Momentum in 2-Dimensions
Data and Calculation Summary
Mass in Kg
m1 = .0162
m2 = .0162
∆ Height (m)
Time Of Flight (s)
∆t = (2∆h/g)1/2
R10 =
R1f =
R2f =
Velocity (m/s)
V0 =
V1 =
V1x =
V1y =
V2 =
V2x =
V2y =
(m)
(m)
(m)
θ1 =
θ2 =
Momentum (Kg∙m/s)
P0 =
P1 =
P1x =
P1y =
P2 =
P2x =
P2y =
Using the information summarized in the table above complete the following:
1. Looking at conservation of momentum in the x-direction, compare the initial momentum to
the final momentum. See the equation (1) in the Background section. Was momentum
conserved?
Equation (1)
m1vo = m1v1cosθ1 + m2v2cosθ2
2. Calculate the percent difference between the initial x-momentum and the total final xmomentum.
percent difference 
p fx  p 0
p fx  p 0
 100
2
3. Consider conservation of momentum in the y-direction. Since there was not initial
momentum in the y-direction, it may be more meaningful to compare the final momentum in
the (+y) direction to the final momentum in the (–y) direction. See the equation (2) in the
Background section. Was momentum conserved?
Equation (2)
m1v1sinθ1 = -(m2v2sinθ2)
4. Since there was no initial momentum in the y-direction, calculate the percent difference
between the final momentum in the (+y) direction and the final momentum in the (-y)
direction.
percent difference 
p (  y)  p (-y)
p (  y)  p (-y)
 100
2
5. For an off center collision between a stationary target and a moving projectile to be
considered perfectly elastic, the sum of the angles for the final projectiles should equal 90°.
According to this rule, is your collision nearly elastic?
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