The Conservation of Momentum©98
Experiment 6
Objective: To verify the law of the conservation of momentum for rectilinear motion.
DISCUSSION:
The law of the conservation of momentum is the basic law of physics. The
momentum of a body with mass m and velocity v is defined to be mv. Since velocity is a
vector and mass a scalar, momentum is a vector.
Newton's first law defines inertial frames and the second law defines force, which
is a measure of the interaction between two bodies. Newton's third law (the law of the
conservation of momentum), puts limits on the consequences of these interactions.
Specifically, the law states that if two bodies, A and B, interact with one another, the
force that A experiences due to B is equal and opposite to the force that B experiences
due to A. As a result it can be shown that the total momentum of a system of bodies
which interact only with one another remains constant in time.
In the special case in which only two bodies, 1 and 2, interact with one another,
Newton's third law can be written as:
(1)
m1 v1b  m 2 v 2b  m1 v1a  m 2 v 2a
Here the subscripts 1 and 2 identify the bodies and the subscripts b and a refer to the
velocities before and after the interaction.
Equation (1) is a vector equation. Thus there is a separate equation for each of the
three perpendicular directions of space. However, in this laboratory we consider only
motion along a straight line, and so only the magnitudes of the velocities and their
directions left or right need be measured.
We return to the air-track to produce horizontal motion which is very nearly free
of outside forces, such as friction with the track and resistance from air. Two carts
moving along the track are allowed to collide with one another forming an isolated, one
dimensional system, so far as the limited horizontal motion provided by the track is
concerned. If the speeds and directions of the two bodies are measured immediately
before they collide and then measured immediately afterward, we expect to find good
agreement with Eq. (1). A second and somewhat restricted conservation law may apply
to the collision. It may be that kinetic energy is conserved in the interaction. For this
case we write that
1
1
1
1
m1v12b  m 2 v 22b  m1v12a  m 2 v 22a
(2)
2
2
2
2
Momentum is always conserved in all situations (provided all interacting bodies
are included in the equation expressing the conservation of momentum). However, the
6-1
total kinetic energy of the system may not be conserved. Changes in the kinetic energy of
the system may appear as compensating changes in other forms of energy associated with
these interactions, such as potential, chemical, electromagnetic, and thermal energy.
Interactions which conserve the total kinetic energy of the system are called
perfectly elastic. Interactions in which the bodies ultimately all have identical velocities,
for example, if they stick together on colliding, are called inelastic. The perfectly elastic
interaction occurs only at the atomic level, but it may be approximated in the laboratory
with two bodies colliding on the air-track provided they are equipped with springy
bumpers. On the other hand, the bodies may also collide completely inelastically if they
stick together on contact.
EXERCISES FOR ELASTIC COLLISIONS:
Apparatus setup: We will use two photogates and a level airtrack to measure the
velocity of two frictionless carts before and after elastic and inelastic collisions. The
momentum of the system before the collision will then be compared to the momentum
after. A similar comparison of kinetic energy will also be conducted.
1. Determine the mass of each cart using the scale provided.
2. Setup the Signal Interface and photogates as shown in Fig. 1.
a. Plug the power brick into the back of the Interface, then into the strip on your
table.
b. Connect the white cable into the back of the Interface then into your
computer’s USB port.
c. Plug a photogate into Channel 1. This will be photogate 1.
d. Plug a photogate into Channel 2. This will be photogate 2.
e. Turn on the Interface (switch is on the back).
Figure 1: PASCO Interface setup.
6-2
3. Start DataStudio. It should be located somewhere under your start menu.
4. When prompted, choose ‘Create Experiment’.
5. Set up the software to recognize the hardware:
a. In the software, add a “Photogate” to Channel 1.
b. Repeat for Channel 2.
5. Set up the constants:
a. Find the “Constants” menu for the first photogate. In version 1.9.7r8, this tab is in
the menu that is seen when you click on the photogate icon shown connected to
the interface.
b. Enter the width, in meters, of the motion (white) tab found on the cart.
c. Repeat for the photogate icon in Channel 2.
d. Find “Photogate Spacing” (in version 1.9.7r8, this is under the Constants tab of
the second photogate) and enter the length, in meters, between the two photogates.
6. Set up the tables.
a. Notice that there is a list of Data options on the upper left (usually). Also notice
that there is a list of Display options on the bottom left (usually).
b. Click and drag the “Table” icon and drop it on “Velocity in Gate, Ch 1”.
c. Click and drag the “Table” icon and drop it on “Velocity in Gate, Ch 2”.
d. Move and resize the table so you can see everything.
7. Test your equipment:
a. Turn on the air supply and test to see that the track is level, or nearly level, by
noting whether a cart initially at rest in the middle of the track remains at rest.
Adjust the leveling support screw at one end of the track, if necessary.
b. Set the smaller of the two carts in the middle of the photogates, so that it remains
at rest.
c. Set the larger of the two carts at one end and push it towards the middle so that it
hits the smaller carts elastically.
d. Observe how the carts move through the gates; for example, the larger cart might
cut the first photogate, then hit the smaller cart and stop completely while the
smaller cart continues through the second photogate.
6-3
Data Acquisition: For elastic collisions, cart 1 (the larger cart) will travel through the
first gate and hit cart 2 (the smaller cart) at rest between the photogates. You WILL have
two velocities for one gate and one velocity for the other gate. It may take some practice
to get things just right; remember to catch the carts before they rebound off the end
bumpers!
1. Click on the ‘Start’ button in the main window to record data.
2. Put cart 1 between the photogates and have it be at rest.
3. Put the cart 2 initially outside of photogate 1 and give it a push towards cart 1.
4. Allow the carts to collide and pick them up off the track after they have passed
through the respective photogates.
5. Repeat the steps 1-4 several times.
6. Click on ‘Stop’ after you catch the carts the last time.
7. Save the data for elastic collisions.
8. Note that one gate has twice as much data as the other. This is because one gate timed
two carts passing through. Remember which cart passed though first and match the
data!
Data Analysis: We can now compare the momentum and kinetic energies before and
after the elastic collision. In Excel:
1. Use the velocity data from the photogates to calculate the momentum of the carts
before and after the collision.
2. Use the velocity data from the photogates to calculate kinetic energy of the carts
before and after the collision.
3. Calculate the percent difference between the momentum before and after the
collision.
4. Also, calculate the percent difference between the kinetic energy before and after the
collision.
5. Questions:
a. Was the momentum of the system conserved? Why or why not?
b. Was the kinetic energy of the system conserved? Why or why not?
6-4
EXERCISES FOR INELASTIC COLLISIONS:
1. Make sure you have saved the elastic collision data.
2. Remove the motion tag (white plastic tag) from the smaller cart (cart 2).
3. Under the “Experiment” menu, choose “Delete all Data Runs”.
Data Acquisition: For inelastic collisions, cart 1 will travel through the first gate and hit
cart 2 (at rest between the photogates). After the collision, both carts will stick together
and move through the second gate. It may take some practice to get things just right;
remember to set the carts with the Velcro facing each other and to catch the carts before
they rebound off the end bumpers!
1. Click on the ‘Start’ button in the main window to record data.
2. Put cart 1 between the photogates and have it be at rest.
3. Put the cart 2 initially outside of photogate 1 and give it a push towards cart 1.
4. Allow the carts to collide and stick together.
5. Stop them after they have passed through photogate 2 but before they bounce back.
6. Repeat the steps 1-5 several times.
7. Click on ‘Stop’ after you catch the carts the final time.
8. Save the data for inelastic collisions.
Data Analysis: We can now compare the momentum and kinetic energies before and
after the elastic collision. In Excel:
6. Use the velocity data from the photogates to calculate the momentum of the carts
before and after the collision.
7. Use the velocity data from the photogates to calculate kinetic energy of the carts
before and after the collision.
8. Calculate the percent difference between the momentum before and after the
collision.
9. Also, calculate the percent difference between the kinetic energy before and after the
collision.
6-5
10. Questions:
a. Was the momentum of the system conserved? Why or why not?
b. Was the kinetic energy of the system conserved? Why or why not?
6-6