Jason Fong
702847140
date of experiment: 10-31-00
partner: Rosanne Hong
Physics 4AL Lab 5
Experiment 4
Abstract and Introduction
The purpose of this experiment was to investigate the general form of Newton's Second


dP
Law: F 
. This investigation involved the impulse on a glider during a collision. The
dt
impulse obtained from the measurements of a force transducer were compared to the impulse
calculated from the velocities before and after the collision. The two methods for calculating the
impulse gave values that agreed to within approximately 1%-2%. The experiment was
performed by pushing a glider down an air track to collide with a force transducer at the end of
the track. The force measurements from the force transducer were graphed against time, and the
integral of the resulting curve gave the impulse of the collision. A photogate was position near
the end of the track so that it could be used to measure the time the glider took to travel a known
distance. The velocity of the glider before and after the collision could then be determined.
From those velocities, the momentums could be calculated, and the impulse could be determined
from the change in those calculated momentums. Then the impulses obtained from the two
different methods could be compared.
Formulas
In order to calculate the values for this experiment, a few formulas needed to be derived.
The first is a formula to convert from the voltage measurements of the force transducer to a force
in Newtons:
F  KV
where F is the force, K is the conversion factor from voltage readings to a force in Newtons, and
V is the voltage measured by the force transducer. The procedure to obtain the value of K is
detailed in the following procedure section. The uncertainty in F is given by:
F  VK
The value of the impulse can be determined using this formula by finding the integral of the
force over the small time intervals measured by the force transducer:
J   Fdt
This is the area under the first large bump of the graph of the force versus time.
The velocities of the glider before and after the collision can also be used to find the
impulse. The formula for the momentum of the glider is given by:
P  mv  m
d
t
where P is the momentum of the glider, m is the mass of the glider, v is the velocity of the glider,
d is a known distance that the glider will travel, and t is the time taken to travel that distance.
The uncertainty in the momentum calculated in this manner is given by:
2
2
d
 m
  md 
P   m    d    2 t 
t
 t
  t

2
the impulse is given by this formula:
J  P2  P1  m2
d2
d
 m1 1
t2
t1
since the glider is the same for both momentums:
 1 1
J  md   
 t2 t1 
where J is the impulse, m is the mass of the glider, d is a known distance the glider will travel,
and t1 and t2 are the times taken for the glider to travel across that distance before and after the
collision. The uncertainty in the impulse calculated in this manner is given by:
2
2
  1 1     1 1    md
  md

J   d   m    m  d    2 t2    2 t1 
  t1

  t2 t1     t2 t1    t2
2
2
Procedure
In order to determine the force exerted on the force transducer, a constant K needed to be
found to convert from the voltage readings to the force exerted on the force transducer. This
was accomplished by hanging weights of various masses on the force transducer. The voltage
readings from the force transducer were taken and the force of the weights pulling on the force
transducer were graphed against the voltage measured. A line was fitted to the resulting scatter
plot, and the slope of the line gave the value for K. The value of K obtained in this manner was
9.44 ± 0.04 N/V. The units of K are Newtons/Volts, which gives the conversion to Newtons
from Volts. The intercept of the line does not factor into the slope and the conversion factor; but
even so, it would not be correct to force the line to intercept the vertical axis at zero since it
would change the position of the line with respect to the points of the scatter plot. The intercept
does not appear in the value for the slope, but changing the intercept will change the value of the
slope.
To prepare for the experiment, the air track was leveled by adjusting the feet of the track
and checking that the glider did not move when the air was turned on. If the track were not level,
it would cause the glider to accelerate toward or away from the collision with the force
transducer, depending on which way the slant of the track was oriented. This would affect the
value of the impulse calculated from the velocities before and after the collision because the
glider would change velocities during the time it travels between the photogate and the point of
collision with the force transducer.
The experiment was set up by placing a glider on an air track. The glider had a solid flag
attached to its top, and the length of the flag was measured to be 3.80 ± 0.05 centimeters. The
mass of the glider and flag was measured to be 201.1 grams. A force transducer was positioned
at one end of the track so that the glider would strike it and reverse direction. The force
transducer was set up to take 2000 readings per second. A photogate was placed near the end of
the track so that the flag of the glider would travel through the photogate just before the glider
struck the force transducer. The photogate would be used to measure the time that the glider
took to travel a distance equal to the length of the flag. The time could then be used to calculate
the velocity of the glider before and after the collision. Those velocities could be used to find the
momentums before and after the collision. The difference in those momentums gives the
impulse.
The glider was pushed towards the force transducer, and when the glider's flag first
blocked the photogate, data recording began. The force transducer took readings for the next 0.5
second, and the photogate recorded the time that the glider's flag took to travel across its beam.
The photogate recorded both the times for the glider's approach towards the force transducer, and
the glider's movement away from the force transducer after the collision.
Data Analysis and Error Analysis
The measurements from the force transducer were converted to force in Newtons using
the conversion factor obtained above. The forces were then graphed against time. The values of
the graph before the collision gave the baseline reading of the force transducer, so the values of
the graph were shifted so that the baseline would lie at zero on the graph. The impulse of the
collision could be found by taking the integral of the area below the first large bump of the
graph. This is because J   Fdt . The vertical segments of the graph are the force F and the
horizontal increments are dt. Only the first large bump is used in the integral because the rest of
the oscillations in the graph are due to the vibrations of the force transducer after the collision.
Only the first large bump represents the force of the collision of the glider. The integral is found
by computing the Riemann sum for the region. The small time intervals of the force transducer
measurements were taken as the intervals for the Riemann sum. The values of the force at each
time interval are multiplied by the 0.0005 seconds time interval and then are added together to
obtain the Riemann sum. The time interval is 0.0005 seconds since there are 2000 readings
taken per second, and 1/2000 is 0.0005. The values of the impulse (JT) obtained in this manner
for each of the five runs is given in the table below. Also included in the table are the values
calculated from the velocities: the momentum before the collision (P1) and after the collision
(P2), the value of the impulse (JP) calculated from those momentums, and the absolute and
percent difference of that impulse from the impulse obtained from the force transducer
measurements.
Force Transducer
Run #
J T   Fdt
(N•s)
Photogate
P1
P2
JP
(N•s)
(N•s)
(N•s)
JP Difference from JT
Absolute
Percentage
(N•s)
1
0.371
0.196 ± 0.003
-0.170 ± 0.002
0.366 ± 0.001
0.005
1.47%
2
0.393
0.207 ± 0.003
-0.178 ± 0.002
0.384 ± 0.001
0.008
2.16%
3
0.420
0.225 ± 0.003
-0.191 ± 0.003
0.416 ± 0.001
0.004
1.07%
4
0.350
0.186 ± 0.002
-0.156 ± 0.002
0.342 ± 0.001
0.007
2.13%
5
0.398
0.212 ± 0.003
-0.182 ± 0.002
0.394 ± 0.001
0.004
1.02%
The values for the absolute differences were calculated using J P  JT , and the values for the
percentage differences were calculated using
J P  JT
. The uncertainties are based on the
JT
previously quoted uncertainties and on an uncertainty of 0.0001 seconds for time.
The low percentage error shows that the value for the impulse obtained through the force
transducer and through the velocities are in fairly close agreement. The slight different can be
attributed to a few sources of error in the experiment. The glider loses some velocity as it travels
along the track due to friction. Thus, the velocities measured at the photogates are not exactly
the same as the velocities just before and just after the collision. Since the velocity changes
slightly between the photogate and the transducer, the momentum also changes. The photogate
was placed close to the transducer in order to minimize this error. If the photogate was very far
from the transducer, then the error from the difference in velocity due to friction would be
greater. During the collision, there is also some energy lost to sound and heat. The rubber
covered tip of the glider also absorbed some of the energy of the collision, which makes the
collision inelastic. Even though energy is not conserved in the collision, momentum is still
mostly conserved since the track did not move very much from the collision so that the glider
still had most of the motion. The momentum difference should still be equal to the impulse
measured by the force transducer.
Conclusion
The values for the impulse calculated from the measurements from the force transducer
and the measurements from the photogate agreed fairly well. They came within approximately
1%-2% of each other. The results of this experiment suggests that momentum is conserved in
the system since the momentums before and after the collision are fairly close. The loss of
momentum can be attributed to friction on the air track and the rubber covering over the tip of
the glider.