D.E. Shaw
©illanova University
January 2, 2007
(Edited September 29, 2015, M. DeGeorge)
Lab #6 - Conservation of Momentum
and
Impulse-Momentum Principle - 850
Spring 2016
Construct a plot of the distance versus time.
Place cart A about 0.20 (m) from the motion sensor II and
Cart B about 0.20 (m) from cart A as shown in the diagram.
The Velcrotm fasteners on the carts should be facing each
other so that the carts will stick together when a collision
occurs. Cart A is given a push towards cart B and the motion
sensor II records the position of A before the collision and
both carts after the collision.
Start collecting data, then give cart A a gentle push
towards cart B. The carts must stick together after the
Equipment: dual range Pasco motion sensor II, Pasco force
collision. Examine the distance versus time plot. The
sensor with bracket and collision bumpers, Pasco Universal
distance should increase linearly with time before the
Table Clamp, two carts, Pasco Cart Track, a set of masses
collision and the slope of this part of the graph gives the
with hanger and the Capstone program.
speed of cart A before the collision. After the collision the
distance should also increase linearly with time but with a
Part A: Conservation of Momentum (Totally Inelastic
different slope that is the combined speed of both carts. A
Collision)
typical data set is shown in Fig. 2. If the data are not
relatively smooth try changing the alignment of the motion
Theory: Suppose two objects are considered as a system in
sensor II, its wide/narrow angle
which the objects exert equal but
cart B
motion sensor II cart A
setting and/or reducing the sampling
opposite forces on each other and the
speed. In addition to the speed
net external force on the system is zero.
change caused by the collision, the
Under these conditions the impulses
carts will slow down from their initial
associated with the internal forces
ds
Al. track
velocity due to resistive forces. If you
change the momentum of the individual
obtain the linear fit for all the data
objects but the total momentum of the
Equipment for Part A
before the collision, the measured
system cannot change. In the totally
Fig. 1
velocity will likely be larger than the
inelastic collision of the two objects the
velocity just before the collision due to the various sources
total momentum must be conserved and the total momentum
of friction. Similarly if you fit all of the data after the
just before the collision must equal the total momentum just
collision you will obtain a final velocity, which is too small.
after the collision. The total kinetic energy of the carts after
the collision is less than the total kinetic energy of the carts
Therefore using the
icon, located above the graph, do a
before the collision.
linear fit for the data just before the collision (these points
are shown in a box in the figure) and for a set of points just
Data Collection: The equipment used to study momentum
after the collision (also shown in the figure). The points
conservation is shown in Fig. 1.
nearest the collision provide a closer
Measure the mass of both carts. If the masses
estimate of the initial and final velocities
1
distance (m)
of the carts are not the same mark the carts with
just before and after the instant of collision.
collision
a small piece of tape to distinguish them.
0.8
Record the slopes (velocities) and the
 Connect the wires from the motion sensor II to
masses of both carts. A single or multiple
0.6
the digital inputs 1 and 2. The wire with the
data highlight box(es) can be used but the
yellow tape (or with the yellow plug) goes to
0.4
slope will only display in one box at a time.
channel 1. Start with the switch on the motion
Do a total of 3 trials using different initial
0.2
sensor II in the wide angle position (you may
speeds.
have to change this to get the best data).
0
Place a 0.5 kg (500 gram) mass on top of
 Click on the Hardware Setup. A window
0
0.5
1
1.5
2
cart A and do three trials with different
time (s)
showing the available sensors will open. Type
initial speeds. Record the slopes and masses
Collision
of
Two
Carts
the letter m and select the motion sensor II in
of the carts (including the added mass) in
Figure 2
this list. Set the sample rate to 50 (Hz).
kilograms.
Introduction: The first objective of this experiment is to
investigate the Conservation of Momentum Principle by
examining a totally inelastic collision of two carts. The
second objective is to study an almost elastic collision and
apply the Impulse - Momentum Principle to one of the
objects. A final objective is to observe how the maximum
force exerted on a cart colliding with a fixed object is related
to the duration of the collision.
1
Place the 0.5 kg mass on top of cart B and do three trials
with different initial speeds. Record the slopes and masses of
the carts (including the added mass) in kilograms.
Analysis:
Open an Excel worksheet and enter the masses of each cart
(including any added mass) and the initial and final speeds.
 Compute the total momentum before and after the
collision. Also compute the total kinetic energy before and
after the collision.
 Find the percent change in momentum 100*(pf - pi)/pi.
In Excel plot a graph of the total momentum of both carts
after the collision versus the total momentum before the
collision using all of the data. Do a linear fit through the
origin.
The Theoretical Fractional Change (TFC) and the
Experimental Fractional Change (EFC) in the kinetic energy
of the system, caused by the totally inelastic collision, is
easily computed by using conservation of momentum to find
the final speed and is:
The model assumes that the spring and the force sensor are
ideal so that no kinetic energy is transferred into the spring
as “internal energy”. During the collision kinetic energy of
the cart is converted into elastic potential energy stored in
the spring and the flexible bar of the force sensor. For the
ideal spring this energy is completely returned as kinetic
energy after the collision. Therefore, for this elastic
collision model the kinetic energy does not change:
1 2 1 2 1
mvi  mv f  MV 2
2
2
2
...2
The speeds after the collision may be found by substituting
Eq. (1) into (2) and solving.
mM 
vf  
vi ...4
mM 
 2m 
V 
vi ...5
mM 
EFC = TFC
𝐾𝑓 − 𝐾𝑖
𝑚𝐵
=−
𝐾𝑖
𝑚𝐴 + 𝑚𝐵
Based on the calculations and the graph and allowing for
reasonable experimental errors, was momentum conserved
during the totally inelastic collision? Explain carefully.
Plot a graph of the experimental fractional energy change
versus the theoretical fractional energy change. Do a linear
fit through the origin. Comment on the significance of this
graph.
Part B: Conservation of Momentum, Energy and the
Impulse – Momentum Principle Applied to an Almost
Elastic Collision.
Since “M” is much larger than “m” the elastic collision
model predicts that:
vf 
V
lim  m  M

M  m  m  M

v i   v i

...6
lim  2m 

vi  0 ...7 
M  m  m  M 
According to the elastic model the cart should be moving in
the opposite direction with the same speed it had just before
the collision provided that the force sensor is securely
fastened to the table so that the mass “M” is very large.
The Impulse-Momentum Principle: The impulse of a force
is the integral of the force over a specific time interval. The
impulse momentum principle states that the change in
Theory:
momentum of the object during this time interval equals the
Momentum and Energy Conservation: An almost elastic
impulse of the net applied force. This principle may be
collision is studied by having the cart of mass “m” collide
tested by measuring the momentum change of the cart
with a spring that is attached to the force sensor which is
shown in Fig. 3. The force sensor exerts a force on the cart
clamped firmly to the table as shown in Fig. 3. The mass
during the collision time. This force produces an impulse
“M” which collides with the cart not only includes the force
that causes the momentum of the cart to
sensor but also the table and
force sensor
Pasco
cart
motion sensor II
change as a result of the collision.
floor to which the table is
clamp
However, the momentum of the cart does
attached. Therefore the mass
m
M
change even if the speed does not change
“M” is very much larger than
since the directions of the velocity vector and
spring
“m”. Take the positive
ds
plank
momentum vectors change as a result of the
direction to be to the right and
Equipment for Part B
collision. For the real spring not all of the
let “vi” and “vf” be the speeds of
Fig. 3
stored elastic potential energy is returned to
the cart just before and just after
the cart so the speeds before and after the
the collision and “V” be the
collision will not be exactly the same. The force sensor
speed of “M” just after the collision. Applying the
measures the force exerted on the cart by the spring during
Conservation of Momentum Principle:
the collision. By integrating this force over the collision
m vi  v f
mvi  mv f  MV  V 
...1
M


2
time we can find the impulse that is then compared with the
change in momentum of the cart.
Experimental Details and Analysis:
 The Pasco Universal Table Clamp is used to connect the
force sensor to the table at one end of the track to prevent the
force sensor from moving during the collision. The cart is
initially given a speed towards the force sensor, a collision
occurs and the cart then moves in the opposite direction. The
force sensor measures the force exerted on it by the cart
during the collision and the motion sensor II measures the
cart's velocity just before and just after the collision.
The direction of the positive X axis may be chosen to the
right in Fig. 3. If the mass of the cart is “m” and the speeds
before and after the collision are “vi” and “vf”, the change in
momentum of the cart caused by the collision is:

 
 

p  p after  p before  m  v f iˆ  m  vi iˆ

p  m| v f |  | vi |iˆ ...8

Note: vf and vi in this formula are both positive since they
are the magnitudes of the velocities. It is important to note
that the momentum of the cart itself is definitely not
conserved since its momentum change is not zero. By
Newton's III Law the force exerted on the cart by the
transducer is opposite to the force exerted on the transducer
by the cart. If the force exerted on the cart by the transducer
during the collision has a magnitude Ft then the magnitude
of the impulse associated with this force is:
t2
P  I   FT t dt ...9
t1
where t2-t1 is the time interval of the collision. The normal
and gravitational forces exerted on the cart cancel and
therefore produce no net impulse. The impulse momentum
theorem states that the impulse of the force exerted on the
cart is equal to the cart's change in momentum.
 Forces to be measured are applied to the force sensor's
hook or to other adapters such as springs that can be used in
place of the hook. The hook or other adapter is screwed
directly into an internal metal strip that bends a little when a
force is applied. The length of thin wires attached to the strip
change slightly in response to the bending of the strip. The
length change alters the cross sectional area and length of
the wires changing their electrical resistance. The small
change in resistance is ultimately converted to a voltage
output that is linear with the applied force over a reasonably
large range of applied forces. The force sensor is connected
to the Pasco interface box via analog input A. The Capstone
program, records the position of the cart with the motion
sensor II and simultaneously records the force exerted on the
force sensor by the colliding cart.
 Connect the force sensor to channel A. Click on the icon
for Analog Channel A type the letter f and select the force
sensor (not the student force sensor) from the list of
available analog sensors and set the sample rate to 1.00 kHz.
Leave the motion sensor II connected to the digital channels
#1 and #2. Under “Recording Conditions” change the data
collection time to 10 seconds.
 In previous experiments forces were applied that pulled on
the sensor. However, in this experiment forces push on the
sensor and consequently it produces negative values if the
same calibration method is used as before. By Newton’s III
Law the magnitudes of the force exerted by the cart on the
sensor and the force exerted on the cart by the sensor are
equal in magnitude and opposite in direction. Since the
direction of the force on the cart is in the negative “X”
direction the negative force reading returned by the sensor
may be used to represent the force on the cart.
Calibration of the Force Sensor:
In the following steps, the force sensor must be held at
rest in a vertical position (rest it on the edge of the lab
bench).
Click the Calibration icon and in step #1 make sure the
force sensor is selected then click next.
In step #2 insure that the force measurement is selected
and press next.
In step #3 select the two standards (2 point) calibration
and hit next.
 In step #4, with nothing on the sensor, momentarily press
the TARE button on the sensor. Enter 0.00 in the Standard
value box and click on the “Set Current Value to Standard
Value” button. Click next.
In step #5 add an additional mass of 1.00(kg) to the mass
hanger for a total mass of 1.05 kg (corresponding to a total
gravitational force of 10.29 N). Attach the hanger and mass
to the force sensor. Enter the value of 10.29 in the
Standard Value Box and click on the “Set Current Value to
Standard Value” button. Click next then finish if the results
are satisfactory.
It is important to verify that the force sensor is working
correctly. Holding the force sensor vertically against the
bench with nothing connected to it, press the TARE button.
Connect the hanger and the added mass to the force sensor
and collect data for a few seconds and then carefully remove
the mass and hanger while still collecting data. Make a plot
of Force vs time. The graph should indicate a net force of
zero while nothing was connected and 10.29 (N) with the
hanger and added mass connected. If these values are not
obtained the calibration process must be repeated. Include a
copy of the calibration plot with the lab report.
 For this part of the experiment use the spring adapter with
the smaller force constant (weak spring) rather than the
hook.
 Under “Recording Conditions” select a new data collection
time of 2.0 seconds.
 The following operation will take a little practice since a
significant time delay occurs between your clicking on the
Start icon and the commencement of data collection. Give
3
Using the
icon, select a range of data points before the
collision and obtain the linear fit and record the slope which
is the magnitude of the velocity before the collision. In the
same way determine the magnitude of the velocity after the
collision. In order to minimize errors due to resistive forces,
use a reasonably small range of data points before and
after the collision as was done in Part A. The slope is
negative after the collision but just record the absolute value
of this slope since only the magnitudes of the velocities in
Equation (8) will be used.
 The impulse is obtained by finding the area under the force
curve during the collision. Because the force applied to the
force sensor is a push rather than the pull used for calibration
the force curve will be negative. Since the direction of the
force exerted by the force sensor on the cart is in the
negative “X” direction the force should be negative. The
calibration that was performed should give negative values
when the cart hits the force sensor. A typical force curve is
shown in Fig. 5. Ideally the base line for the force versus
time graph should be at zero. However, it is possible that a
drift has changed the base line slightly. If the base line is not
close to zero, significant errors will occur when the
integration under the force curve is done. If the base line is
not very close to zero use the Calculator to create a new
corrected force by adding or subtracting the required amount
to make the baseline zero. Click on the Calculate button and
add a new equation. Change the function name to Fc. Right
click to the right of the equals sign and select Insert Data
then choose Force. Next add or subtract the offset from zero
in the plot. The calculator window should display:
"Fc = [Force(N)] - .01" assuming the correction is -0.01.
Click on the Units box and
Time (s)
0
type n then choose N. Click
1
1.05
1.1
1.15
on Accept to enter the
-1
corrected force. Make a new
-2
plot of the corrected Force
-3
(Fc) versus time.
 An examination of the
-4
Force versus time graph
-5
reveals that the collision time
-6
is very short. Select the
region where F is non zero
1.2
Force (N)
the cart a gentle push towards the force sensor and click on
the Start icon before the collision occurs. Even though the
force sensor is clamped you should use your hand to
provide extra support to prevent the force sensor from
moving during the collision. The objective is to obtain
about the same number of data points before and after the
collision. If your timing is good you should see a triangular
plot for the distance as shown in Figure 4. Try a few more
runs until the plot is similar
0.8
to Figure 4. The linear
distance (m)
range of the force sensor is
0.6
from 0 to 40 Newtons. A
larger force could possible
0.4
collision
damage the sensor.
Therefore it is important
0.2
to limit the maximum
applied force to 40 N.
0
Actually for reasonable
0
0.5
1
1.5
speeds it will probably be
time (s)
much less than this value in
Fig. 4
Collision with Force Sensor
this part of the experiment.
Keep practicing until a
satisfactory data set has been obtained. Check with the
instructor if you are not sure.
 The velocity before the collision can be found from the
slope of the linear fit of the distance versus time graph.
Fig. 5
2
and use the Area
icon
to obtain the integral of the force over time, or impulse. The
precision of the area measurement may be increased by
clicking on the Data Summary icon on the left of the screen
and clicking on
under the Force Sensor Bar
and choosing the gear icon. In the Gear Icon dropdown
select the Numerical Format. Change the precision to
something like 3 or 4 if necessary. Use the Show
Coordinates tool
to measure the time t1 at the
beginning of the collision and the time t2 at the end of the
collision.
 Do at least six more trials, in each case varying the initial
speed of the cart. Press the Tare button before each trial. For
each trial be sure to record the initial velocity, final velocity,
the times t1 and t2 and the impulse.
 All of the following calculations can be done conveniently
in Excel. You can use the keys ALT + TAB to switch back
and forth between Capstone and Excel. Transfer the
velocities before and after the collision and the impulses
values to Excel. The impulse of the force exerted by the
force sensor on the cart is negative.
 The model predicted that the speed just before and after the
collision should be the same and therefore the kinetic energy
of the cart should not change.
 For each trial compute the kinetic energy of the cart just
before and just after the collision. Is the collision elastic?
Explain clearly why the energies may not be exactly the
same.
Plot a correlation graph of the kinetic energy after the
collision versus the kinetic energy before the collision and
find its slope. How well are these quantities correlated?
 Compute the change in momentum of the cart using
Equation (8).
 Compare the net impulse with the change in momentum.
How do they compare? Find the percent difference. Do the
experimental results support the impulse - momentum
theorem?
 Plot a correlation graph of the change in momentum versus
the impulse and find its slope. How well are these quantities
correlated?
Explain clearly why the momentum of the cart itself was
not conserved during the collision. Note that this has nothing
to do with the elastic or non-elastic nature of the collision.
4
For the first trial find the change in momentum of the force
sensor and the table (think about this!).
Assuming an ideal spring determine the energy stored in
the spring when it was compressed by its maximum amount
during the collision. Do this for the first trial only.
For each trial calculate the collision time t = t2 – t1.
Intuition might suggest that the collision time would
increase with the speed of the cart since the maximum spring
compression will increase. However, in the next experiment
we study the motion of a mass connected to a spring and will
find that the cart moves with Simple Harmonic Motion when
it is in contact with the spring. An important property of this
kind of motion is that the compression of the spring is a sine
function of the time. The time (T) for one complete
oscillation of the sine function is the period of the motion
and is independent of the maximum spring compression
which is also the amplitude. The motion shown in Fig. 5
represents one half of a complete oscillation of the sine
function and as a result the period is: T = 2t. The plot
shown in Fig.5 is the force versus time. However, according
to Hooke’s Law the force exerted by a spring is proportional
to its compression and so Fig. 5 also represents the spring
compression versus time and should be one half of a
complete oscillation of the sine function. Make a plot of t
vs initial velocity. Do a linear fit (not through the origin)
and record the correlation. Rescale the vertical axis between
0 and 0.4 sec to minimize the effects of scatter in the data.
Does the collision time, t, seem to be independent of the
speed of the cart before the collision (which determines the
maximum spring compression)? Note: low correlation
means independence of data. A more detailed study of Fig.
5 is described in the Optional Project section.
Part C: Comparison of the Collision of the Cart with the
Spring and Other Objects Connected to the Force
Sensor.
The purpose of this part of the experiment is to compare the
force versus time graphs for collisions with the hook, rubber
stopper, putty and springs. In order to obtain as many values
of the force during the collision as possible the motion
sensor II is not used.
 Use the Recording Conditions button to set the data
collection time to one second. Change the force sensor
sample rate to 2.00 kHz.
 The absolute value of the maximum force that can be
measured correctly by the force sensor is 40 (N). When the
cart collides with the hook mounted in the force sensor, the
maximum force applied to the hook can easily exceed 40 (N)
even if the cart is moving slowly. To achieve a small cart
speed, elevate one end of the ramp slightly and release the
cart from rest from a location not far from the hook. Place
two, stacked 200 gram masses under the end of the
aluminum track opposite the force sensor to make a ramp
with a very small inclination angle.
 Using the hook in the force sensor, press the TARE button,
release the cart from a distance of less than 40 centimeters
from the hook and begin collecting data just before the
collision. As mentioned, the collision with the hook
produces large peak forces, therefore press down firmly on
the force sensor during the collision to minimize any
possible movement of the sensor during the collision. If the
absolute value of the force is greater than 40N, move the cart
closer to the force sensor and try again. If it is too small
move the cart a little farther from to the hook. The exact
distance is not important just try to use the same distance to
be consistent.
 Change the limits of the time axis to display a range of
about 0.2 seconds centered on the actual collision. Obtain
the area under the curve during the collision, the maximum
force, the average force (mean must be selected in the
statistics option list) and the approximate duration of the
collision. Make sure the baseline of the collision is close to
zero.
 Replace the hook by the spring and then obtain data for the
collision of the cart with the spring. The cart should be
released from the same point as before to produce the
same momentum before the collision.
 Plot this data using the same range for the force axis and
the same time range of 0.2 seconds surrounding the
collision. Obtain the area under the curve during the
collision, the maximum force, the average force and the
approximate duration of the collision.
 Repeat the last two steps for the other spring, the rubber
stopper and a chunk of putty; in each case releasing the cart
from the same initial position used before. Remember to
tare the force sensor before each run.
 Compare the impulses, maximum forces, average forces
and time duration for the collisions with the hook, springs,
rubber stopper and putty.
 The magnitude of the impulse of a force having an average
value Favg applied for a time interval t is:
P  I   Fdt  Favet  Fave 
I
t
...(10)
 Plot the average force versus “I/t” for each type of
collision. Do a linear fit through the origin and compare the
slope with the value predicted by Eq. (10).
Theoretically, the impulse for the collision with the putty
should be half as large as the collision with the spring if the
spring collision is perfectly elastic. Why? How do the
experimental impulses for the putty and spring collisions
compare?
The maximum force exerted on the human body is the
most important factor in determining possible injury in an
automobile accident. Explain how your results for
collisions with different types of objects may be significant
in understanding the effectiveness of airbags in
automobiles.
5