Experiment #8 Conservation of Momentum Pre-lab
Questions
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Elastic Collision
Consider a head-on collision between two carts. One is initially at rest and the
other moves toward it. The carts bounce off each other in an elastic collision.
Figure 1: Experimental Data Obtained from Logger Pro for an Elastic Collision
** IT IS IMPERATIVE THAT YOU UNDERSTAND WHAT THE GRAPH
ABOVE MEANS BEFORE YOU GET STARTED! ***
Recall back to lab 2 – where we were first playing with the motion sensors (pull your
copy out, or look at the example online if you need a “refresher”). Remember what a
positive slope meant and what a negative slope meant. When we had a positive slope
measured by a sensor, this means the object being measured was moving away from the
sensor. When we had a negative slope measured by a sensor, this means the object being
measured was moving toward the sensor.
Below is the setup for Experiment 8 – Conservation of Momentum. Motion sensor 1 is
measuring the distance from itself to cart 1 (this is the red line in the graph above) and
motion sensor 2 is measuring the distance from itself to cart 2 (this is the blue line in the
graph above). Notice that the two motion sensors are pointing in opposite directions …
This means there will be a sign conflict when the objects are moving in the same
direction! Carefully consider the following in conjunction with your experience from past
labs:
Figure 2: Experimental Setup before the any motion occurs (No velocities)
(Graph between times 0s and 1s)
Figure 3: Experimental operation before the collision (cart 1 is moving to the right and
cart 2 is not moving)
(Graph between times 1s and 1.65s)
In this situation, motion sensor 1 will record cart 1 moving away from it (hence a positive
slope of the red line). The magnitude of the velocity (speed) of cart 1 can be obtained
from the absolute value of the slope of the best fit line and the direction is to the right
(assume this is the +x direction). Motion sensor 2 will record cart 2 as not moving (hence
a zero slope of the blue line).
Figure 4: Experimental operation at the collision
(Graph at instantaneous time 1.65s)
Figure 5: Experimental operation after the collision (cart 1 is not moving and cart 2 is
moving to the right)
(Graph between times 1.65s and 2.2s)
In this situation, motion sensor 2 will record cart 2 moving toward it (hence a negative
slope of the blue line). The magnitude of the velocity (speed) of cart 2 can be obtained
from the absolute value of the slope of the best fit line and the direction is to the right
(again, this is the +x direction). Motion sensor 1 will record cart 1 as not moving (hence a
zero slope of the red line).
Now, this means that when we are determining the velocities from the graph, we must get
BOTH the correct magnitude (from the best fit line) and correct direction (from the
analysis above). This means that if we assume the +x direction is to the right, all
velocities we obtain during this experiment should be positive (regardless of the sign on
the best fit lines), since the velocities are ALL in the same direction.
Fill out the tables below and determine weather or not momentum is conserved.
Mass of Cart 1
0.510 kg
Velocity of
Cart 1 before
Collision
[m/s]
0.3475
Momentum
of Cart 1
before
Collision
[kg m/s]
0.177225
Momentum
of Cart 2
before
Collision
[kg m/s]
0
Velocity of
Cart 2 before
Collision
[m/s]
0
Momentum
of Cart 1
after
Collision
[kg m/s]
0.000221595
Mass of Cart 2
0.498 kg
Velocity of
Cart 1 after
Collision [m/s]
0.0004345
Momentum
of Cart 2
after
Collision
[kg m/s]
0.1781844
Total
Momentum
before
Collision
[kg m/s]
0.177225
Velocity of
Cart 2 after
Collision
[m/s]
0.3578
Total
Momentum
after
Collision
[kg m/s]
0.178405995
Ratio of Total
Momentum
(After/Before)
1.0067
Since the ratio of the total momentum is nearly exactly 1, this means momentum is
conserved, and the theory holds true.
Show your work here
Determining the velocity of Cart 1 before the collision:
Using the best fit lines from Logger Pro, we are given the following equation for Cart 1
(red line) before the collision:
m

position 1   0.3475  t   0.3430 m 
s

If we compare the equation for kinematic 2D motion for the x-position with the found
trendline, we can easily see by evaluation that the following constants correlate directly:
x  vo , x t  xo
m

x   0.3475  t   0.3430 m 
s

Thus:
vo , x [m/s]
0.3475
x o [m]
-0.3430
Determining the velocity of Cart 1 after the collision:
Using the best fit lines from Logger Pro, we are given the following equation for Cart 1
(red line) after the collision:
m

position 1   0.0004345  t  0.2648 m 
s

If we compare the equation for kinematic 2D motion for the x-position with the found
trendline, we can easily see by evaluation that the following constants correlate directly:
x  vo , x t  xo
m

x   0.0004345  t  0.2648 m 
s

Thus:
vo , x [m/s]
0.0004345
x o [m]
0.2648
Determining the velocity of Cart 2 after the collision:
Using the best fit lines from Logger Pro, we are given the following equation for Cart 2
(blue line) after the collision:
m

position 2    0.3578  t  0.6243 m 
s

If we compare the equation for kinematic 2D motion for the x-position with the found
trendline, we can easily see by evaluation that the following constants correlate directly:
x  vo , x t  xo
m

x    0.3578  t  0.6243 m 
s

Thus:
vo , x [m/s]
0.3578
x o [m]
0.6243
** [NOTE: By examining the setup of the system (see comments above), we can see that
the second sensor will record a negative velocity toward the sensor for cart 2. We then
must change its sign to compensate for the direction the cart is actually moving. Recall
that a velocity is both a magnitude and direction – so both must be carefully considered
when analyzing momentum problems.] **
Determining the momentum of Cart 1 before the collision:
We know that momentum is given by:
p  mv
Using the velocity we found from above, we can plug everything right into the equation
for momentum.
m

p1,before  0.510 kg  0.3475   0.177225 N s
s

p1,before  0.177225
kg m
s
Determining the momentum of Cart 1 after the collision:
m

p1,after  0.510 kg  0.0004345   0.000221595 N s
s

p1,after  0.000221595
kg m
s
Determining the momentum of Cart 2 after the collision:
m

p 2,after  0.498 kg  0.3578   0.1781844 N s
s

kg m
s
p 2,after  0.1781844
Determining the total momentum before the collision:
n
pTotal,before   pi ,before  p1,before  p 2,before
i 1
kg m   kg m 

pTotal,before   0.177225
  0

s  
s 

pTotal, before  0.177225
kg m
s
Determining the total momentum after the collision:
n
pTotal,after   pi ,after  p1,after  p 2,after
i 1
kg m  
kg m 

pTotal,after   0.000221595
   0.1781844

s  
s 

pTotal,after  0.178405995
kg m
s
Determining the ratio of total momentums (After/Before):
Ratio of total momentums 
Ratio of total momentums 
pTotal,after
pTotal,before
kg m
s  1.0067
kg m
0.177225
s
0.178405995
Since the ratio of the total momentum is nearly exactly 1, this means momentum is
conserved, and the theory holds true.
Inelastic Collision
Consider a head-on collision between two carts. One is initially at rest and the
other moves toward it. The carts stick to each other in an inelastic collision.
Figure 6: Experimental Data Obtained from Logger Pro for an Inelastic Collision
** AGAIN, MAKE COMPLETELY SURE THAT YOU UNDERSTAND WHAT
THE GRAPH ABOVE MEANS BEFORE YOU GET STARTED! ***
Notice, again, that the two motion sensors are pointing in opposite directions … This
means there will again be a sign conflict when the objects are moving in the same
direction! Carefully consider the following in conjunction with your experience from past
labs:
Figure 7: Experimental Setup before the any motion occurs (No velocities)
(Graph between times 0s and 0.6s)
Figure 8: Experimental operation before the collision (cart 1 is moving to the right and
cart 2 is not moving)
(Graph between times 0.6s and 1.45s)
In this situation, motion sensor 1 will record cart 1 moving away from it (hence a positive
slope of the red line). The magnitude of the velocity (speed) of cart 1 can be obtained
from the absolute value of the slope of the best fit line and the direction is to the right
(assume this is the +x direction). Motion sensor 2 will record cart 2 as not moving (hence
a zero slope of the blue line).
Figure 9: Experimental operation at the collision
(Graph at instantaneous time 1.45s)
Figure 10: Experimental operation after the collision (cart 1 is stuck to cart 2 and both
are moving to the right)
(Graph between times 1.45s and 2.2s)
In this situation, motion sensor 2 will record cart 2 moving toward it (hence a negative
slope of the blue line). The magnitude of the velocity (speed) of cart 2 can be obtained
from the absolute value of the slope of the best fit line and the direction is to the right
(again, this is the +x direction). Motion sensor 1 will record cart 1 moving away from it
(hence a positive slope of the red line). The magnitude of the velocity (speed) of cart 1
can be obtained from the absolute value of the slope of the best fit line and the direction
is also to the right. Since these two velocities are one in the same (since the carts are
stuck together) they should be equal – hence it is appropriate to get the values for the
velocity of the carts by averaging the slopes recorded from sensor 1 and sensor 2.
Again, make sure you get BOTH the correct magnitude (from the best fit line) and correct
direction (from the analysis above). This means that if we assume the +x direction is to
the right, all velocities we obtain during this experiment should be positive (regardless of
the sign on the best fit lines), since the velocities are ALL in the same direction.
Fill out the tables below and determine weather or not momentum is conserved.
Mass of Cart 1
0.510 kg
Velocity of
Cart 1 before
Collision
[m/s]
0.3893
Momentum
of Cart 1
before
Collision
[kg m/s]
0.198543
Momentum
of Cart 2
before
Collision
[kg m/s]
0
Velocity of
Cart 2 before
Collision
[m/s]
0
Momentum
of Cart 1
after
Collision
[kg m/s]
0.0962115
Mass of Cart 2
0.498 kg
Velocity of
Cart 1 after
Collision [m/s]
0.18865
Momentum
of Cart 2
after
Collision
[kg m/s]
0.0939477
Total
Momentum
before
Collision
[kg m/s]
0.198543
Velocity of
Cart 2 after
Collision
[m/s]
0.18865
Total
Momentum
after
Collision
[kg m/s]
0.1901592
Ratio of Total
Momentum
(After/Before)
0.95777
Since the ratio of the total momentum is nearly exactly 1, this means momentum is
conserved, and the theory holds true.
Show your work here
Determining the velocity of Cart 1 before the collision:
Using the best fit lines from Logger Pro, we are given the following equation for Cart 1
(red line) before the collision:
m

position 1   0.3893  t   0.3021 m 
s

If we compare the equation for kinematic 2D motion for the x-position with the found
trendline, we can easily see by evaluation that the following constants correlate directly:
x  vo , x t  xo
m

x   0.3893  t   0.3021 m 
s

Thus:
vo , x [m/s]
0.3893
x o [m]
-0.3021
Determining the velocity of Cart 1 after the collision:
Using the best fit lines from Logger Pro, we are given the following equation for Cart 1
(red line) after the collision:
m

position 1   0.1869  t   0.003447 m 
s

If we compare the equation for kinematic 2D motion for the x-position with the found
trendline, we can easily see by evaluation that the following constants correlate directly:
x  vo , x t  xo
m

x   0.1869  t   0.003447 m 
s

Thus:
vo , x [m/s]
0.1869
x o [m]
-0.003447
Determining the velocity of Cart 2 after the collision:
Using the best fit lines from Logger Pro, we are given the following equation for Cart 2
(blue line) after the collision:
m

position 2    0.1904  t  0.2773 m 
s

If we compare the equation for kinematic 2D motion for the x-position with the found
trendline, we can easily see by evaluation that the following constants correlate directly:
x  vo , x t  xo
m

x    0.1904  t  0.2773 m 
s

Thus:
vo , x [m/s]
0.1904
x o [m]
0.2773
** [NOTE: By examining the setup of the system (see comments above), we can see that
the second sensor will record a negative velocity toward the sensor for cart 2. We then
must change its sign to compensate for the direction the cart is actually moving. Recall
that a velocity is both a magnitude and direction – so both must be carefully considered
when analyzing momentum problems.] **
Determining the average velocity of the two carts after the collision:
v 
v1  v 2

2
0.1869
m
m
 0.1904
s
s  0.18865 m
2
s
Determining the momentum of Cart 1 before the collision:
We know that momentum is given by:
p  mv
Using the velocity we found from above, we can plug everything right into the equation
for momentum.
m

p1,before  0.510 kg  0.3893   0.198543 N s
s

p1,before  0.198543
kg m
s
Determining the momentum of Cart 1 after the collision:
m

p1,after  0.510 kg  0.18865   0.0962115 N s
s

p1,after  0.0962115
kg m
s
Determining the momentum of Cart 2 after the collision:
m

p 2,after  0.498 kg  0.18865   0.0939477 N s
s

p 2,after  0.0939477
kg m
s
Determining the total momentum before the collision:
n
pTotal,before   pi ,before  p1,before  p 2,before
i 1
kg m   kg m 

pTotal,before   0.198543
  0

s  
s 

pTotal,before  0.198543
kg m
s
Determining the total momentum after the collision:
n
pTotal,after   pi ,after  p1,after  p 2,after
i 1
kg m  
kg m 

pTotal,after   0.0962115
   0.0939477

s  
s 

pTotal,after  0.1901592
kg m
s
Determining the ratio of total momentums (After/Before):
Ratio of total momentums 
pTotal,after
pTotal,before
kg m
s  0.95777
Ratio of total momentums 
kg m
0.198543
s
0.1901592
Since the ratio of the total momentum is nearly exactly 1, this means momentum is
conserved, and the theory holds true.