Physics 181 - Summer 2013 - Experiment #5
1
Experiment #5, The Conservation of Momentum in One Dimension
The Ballistic Pendulum
1 Purpose
In this experiment we seek to verify the conservation of momentum in an inelastic collision. We
will do this by making independent measurements that allow us to calculate the momentum
before and just after the collision. We will determine the momentum before the collision by
using the distance traveled along the parabolic trajectory that results from the acceleration due to
gravity. The final momentum will be determined using the conservation of energy. We can only
make measurements before and after the collision. We will use these measurements to infer
conservation on momentum.
2 Introduction
2.1 Conservation of Momentum
If there are no external forces acting on an object in motion Newton tells us that it will keep
going at an unchanged velocity. This is equivalent to the conservation of momentum. Unless
something acts on an object to change its momentum that momentum is a constant. As a
consequence any system of masses that is isolated from external forces will have a net
momentum which is conserved. This conservation of momentum holds for the system as a whole
no matter how the masses interact. In this experiment we will have a system composed of a mass
(ball) shot from a gun and a pendulum in which the ball embeds itself and forming an inelastic
collision. This conservation of momentum can be written as:
Pinitial = Pfinal
The momentum of an object is defined as the product of its mass times its velocity, P = mv. Mass
m is a scalar quantity with magnitude but with no directional properties. Velocity v is a vector
quantity having both a magnitude and a direction. In this experiment, the direction is in one
dimension; the positive x direction. As a result, momentum P is a vector quantity. The direction
of the momentum vector is the same as that of the velocity vector v. The magnitude P of the
momentum vector is the scalar product of the mass times the magnitudevof the velocity
vector, the speed:
P= mv= mv = p
In this write-up, scalar quantities are shown in normal type, while vector quantities are in
boldface. The momentum vector will be given by a bold capital P, and its magnitude by p.
In this experiment, a ball, of mass m, is launched at an angle of zero degrees with respect to the x
axis. Therefore, the initial velocity vector vi of the ball is in the direction of the horizontal x-axis
as indicated in Fig. 2. The initial direction of its momentum, Pi = mvi , is also in the x-direction.
At the time of collision, the ball is caught by the pendulum. Just after this ‘collision’ (and just
before the ball and pendulum system rises), the final direction of the momentum, Pf, of the ball
and pendulum system is too along the x-axis. As both momentum vectors are in this one
dimension, the horizontal or x-axis, we can turn our equations into scaler equations. This allows
us to calculate only the magnitude of the vector which is p = mv.
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Figure 1: The Blackwood ballistic pendulum apparatus with spring gun.
The apparatus for this experiment, shown in Fig. 1, consists of a steel ball and a pendulum.
Because the pendulum is stationary before the collision, its velocity (and thus its momentum) is
zero. Thus the total momentum of the system before collision, pi, is the momentum of the ball,
and therefore, pi is the product of the ball's mass m and its speed vi. Note that vi = vball = vx,i.
pi = mvi
At the time of collision, the ball is caught by the pendulum of mass M. They now share a
common speed vf and a combined mass of (M + m). The final momentum of this system, pf, is
given by:
p f  (M  m)v f
Note that vf = vpendulum + ball = vx,f. After the collision the pendulum swings about its center of
support. At its maximum angular displacement, the center of mass of the system will have risen
through a vertical distance h. At this instant all of its initial kinetic energy has been converted to
potential energy. Applying the principle of the conservation of energy, one obtains:
1
(M + m) v 2f = (M + m)gh
2
where g is the acceleration of gravity in this lab. In this experiment, g = 980.35 cm/sec2.
From the last expression, and solving for vf, the velocity of the ball-pendulum system is:
v f = 2gh
Physics 181 - Summer 2013 - Experiment #5
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Substituting, one obtains:
p f = (M + m) 2gh
The center of mass of the pendulum’s holder, with the ball, will raise a net height h. A pawl on
the bottom of the holder catches in a notch on an angled rack to hold the system at its highest
point. This interaction results in a dissipative force so the height h is reduced. The pendulum too
has a pivot point which allows the pendulum to rotate through the air a short distance to attain h.
It should be noted that the collision, the impulse, takes place very quickly, so there is negligible
rotation of the pendulum at the time of impact. Therefore, we are allowed to ignore the
dissipative force of gravity during the collision process. During this quick inelastic collision, the
pendulum and the ball form an isolated system. Total energy is conserved during the collision,
but not kinetic energy. The lost kinetic energy is the thermal energy, which includes sound,
developed during the collision.
2.2 Projectile Motion
In the second part of this experiment, independent measurements are made to determine the
initial horizontal ball speed vi and initial momentum pi. Fig. 2 shows the apparatus on a table
with the ball at a height y above the floor. The pendulum is swung up out of the way as shown in
Fig. 2. In this configuration, when the ball is launched horizontally, it has an unobstructed path to
follow the trajectory shown in Fig. 2.
After the ball is launched by the spring gun with speed vi, it follows the trajectory approximated
by the curved dotted line.
Figure 2: Pendulum swung up out of the way.
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At the instant the ball is launched horizontally from the spring gun, it begins to fall. While it is
falling, its horizontal speed vi is unaffected by the downward acceleration of gravity. Note that
the vertical downward speed, vy, is zero at the instant the ball leaves the gun. This is why the gun
is leveled horizontally with the x axis. When air resistance is negligible, the vertical and
horizontal motions of a projectile are separable, vx and vy. The net result is the accelerated
vertical motion (falling) combined with the horizontal motion at a constant horizontal speed.
This gives the trajectory depicted above in Fig.2. Distance of fall y and range X measurements
will be used to determine the initial speed of the ball, vi = vx,i.
A ball released from rest will fall vertically with the acceleration of gravity g. The ball will fall
through a distance y in a time t related by:
y=
1 2
2y
gt ; t =
2
g
In this time t, a ball launched with a horizontal speed vi will cover a horizontal distance X to the
point of impact on the floor. This distance is given by:
X = vit = vi
2y
g
The speed of the ball is found to be:
vi = X
g
2y
This yields an initial momentum given as:
pi = mvi= mX
g
2y
3 Experimental Apparatus and Procedure
3.1 Apparatus and Method
The apparatus is the Blackwood ballistic pendulum with a spring gun, as shown in Fig. 1.
In the first part of the experiment, a steel ball mounted on a spring gun is launched horizontally
and caught by a holder on the pendulum. The pendulum then raises a height h. From the
measurement of h, the speed vf of the pendulum-ball system (as the pendulum just starts to
swing) is determined. Knowing this speed and the masses of the ball and the pendulum, the final
momentum, pf, of the system after the collision can be determined.
In the second part of the experiment, the pendulum is swung out of the way so that it does not
interfere with the ball. The ball's speed at launch is determined from a measurement of its initial
Physics 181 - Summer 2013 - Experiment #5
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height y above the floor and the horizontal range X to the point of impact. The momentum of the
ball before the collision is its speed vi times its mass m. The momentum of the stationary
pendulum before the collision with the moving ball is zero as it is not moving. Therefore, the
initial momentum of the ball-pendulum system before the collision is just the momentum of the
ball. Conservation of momentum yields:
mX
g
= (M + m) 2gh
2y
3.2 Procedure
The frame of the Blackwood apparatus must be level and firmly clamped to the table. It is
important that the apparatus remain in the same position until all the measurements have been
completed. So, be careful not to move the table or apparatus during the experiment! The
apparatus should be located near one edge of the table, as shown in Fig. 2.
Caution: Make sure no one is in the path of the steel ball in order to prevent injury to other
students or instructors. Never make any measurements when the ball is cocked in the
launch position.
3.2.1. Determination of the Final Speed of the Pendulum-Projectile (steel ball) System
Immediately after the Collision
1.
Release the pendulum from the rack and allow it to hang freely. Identify the index
pointer marking the center of mass of the pendulum. Carefully measure hi (see
Fig. 1), the height of the index pointer above the base of the apparatus. Make five
metric scale measurements. Estimate the experimental error in measuring hi.
2.
Move the pendulum up onto the rack. Prepare the gun for launching by
momentarily resting the pendulum on the rack while the projectile (ball) is placed
on the end of the spring-gun shaft. Hold the gun with one hand, and with the other
hand push the ball back against the spring until the collar on the rod engages the
trigger. This compresses the spring a definite amount. Due to irregularities in the
launching mechanism, you will find some variation in the speed v. This will lead
to a variation in the value of the height hf (see Fig. 1) the pendulum rises.
3.
Release the pendulum from the rack and allow it to hang freely. When the
pendulum is at rest, launch the ball into the pendulum holder. The pendulum will
swing from its initial position, hi, to the final position, hf, as shown in Fig. 1.
4.
Measure hf, the height of the index pointer. Estimate the error in this metric
measurement. To remove the ball, gently push it out of the holder with your
finger.
5.
Repeat steps 2 to 4 to make ten launchings into the pendulum holder. Each
member of the group should take a turn at measuring hf.
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6.
Each member of the group should measure the mass m of the ball using the pan
balance. Record your estimate of the experimental errors for this procedure.
Record the mass M of the ballistic pendulum (given on the apparatus). This mass
was previously measured using the same type of scale balance.
3.2.2 Determination of the Initial Velocity from Measurements of Range and Fall
1.
Measure the height y from the point where the ball is launched to the floor (See
Fig. 2). Note that y is the distance between the floor and the bottom of the
ball, which is resting on the spring gun shaft in the un-cocked state. Obtain
five values of y and estimate the experimental error in the values of y. Members of
the group should take turns measuring the fall distance y. Use the metric scale
when measuring.
2.
Use the plumb bob to find the point on the floor directly below the launch point of
the ball on the gun (before it is cocked) to the floor.
3.
Position the pendulum, as shown in Fig. 2, so that it will not interfere with the
free flight of the ball. Locate the general area of the ball's impact on the floor by
launching the ball a few times. Several observers should watch to help determine
the
point at which the ball strikes the floor. Tape two pieces of paper on the floor to
cover the general area of the expected impact. Launch the ball once more and note
the visible impact that the ball leaves on the paper. Adjust the paper if necessary.
Place a pencil mark in the center of the impact.
4.
Make ten launchings. Once all launching is complete, measure and record the
range X for each launch. Your TA will instruct you in the technique to make these
measurements. Range X is measured from the point directly below the launch
point to the point of impact of the steel ball on the floor. Tabulate the
measurements of X and estimate the experimental error in the values of X.
5.
When you have completed all of the launchings, confirm that the spring is in the
released position and return the ball to the spring gun.
4 Calculations and Analysis of Data
1.
Calculate the mean (average) values of h, y, and X. You will need to calculate the
standard deviation and standard deviation of the mean, SDM, for hi, hf, y, and X.
These values will be needed for error propagation calculations. (See calculation
sheet at the end of this experiment.) The average value of h is given as:
h = hf - hi
2.
Insert the mean values of the range X and the distance of fall y to obtain the speed
vi of the ball before it collides with the pendulum. Calculate the initial momentum
of the system to obtain the value of pi. (Example calculation 4.425x104 g-cm/s.)
Use scientific notation for this calculation, and keep three numbers after the
Physics 181 - Summer 2013 - Experiment #5
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decimal point for subsequent calculations. You will enter this result (example) as
pi = 4. 425x104 g-cm/s = 4.43 x 104 g-cm/s in your lab report.
3.
Calculate the final momentum, pf, of the system just after the collision by
inserting the mean value of h. (Example calculation 4.375x104 g-cm/s). Use
scientific notation for this calculation, and keep three numbers after the decimal
point for subsequent calculations. You will enter this result (example) as pf =
4.375x104 g-cm/s = 4.38x 104 g-cm/s in your lab report.
4.
Use propagation of error (uncertainty) methods to obtain pi and pf, the
respective errors in pi and pf. The “Appendix on Error Analysis” document at our
Web site has information on propagation of error. Note that the error for pi is
given by:
δ pi = pi
( 1 SDM y )
2
2
y
+(
SDM X
X
2
)
+
( δm )
m
2
B
Similarly for pf :
δ pf = pf
(1
2
SDMhi  SDMhf
2
2
2
) +(
h
δm 2
MP  m B
2
)
Above, MP = mass of the pendulum and mB = mass of the ball. The same type of
scale was previously used to measure the mass of the pendulum. δm 2 in the
above expression is equal to δ
δ
as both MP and mB have the same
uncertainty in the measurement of mass; M = m.
5.
Determine the percent error between pf and pi.
% error =
6.
pf  pi
pi
(100%)
Does pi fall within the interval determined by the limits of your experimental error
(precision), ± pf, for pf, which was determined in part 4? We want to know if the
following is true:
| pf – pi |  pf
This is asking if pi is in the interval of values given by:
(pf - pf, pf + pf).
7.
If question 6 is not true, what type of error, or errors, may account for this?
Explain.
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5 Questions
1.
Is momentum in the horizontal direction conserved in this experiment within 5%?
(Use percent error with the initial momentum as the accepted value.)
2.
State possible sources of error in this experiment (random, systematic and
personal mistakes) and their relative importance.
3.
Kinetic Energy [K.E. = (½) mv2 = p2/2m] Use scientific notation when calculating
kinetic energy. Keep two numbers after the decimal point.
(a)
Calculate the initial kinetic energy (K.E.)i of the system before collision.
(b)
Calculate the final kinetic energy (K.E.)f of the system an instant after
collision.
(c)
Next, calculate the fractional loss of kinetic energy given by:
Fractional loss = [ | (K.E.) f - (K.E.) i | / (K.E.) i ]
4.
(d)
Compute the ratio of the mass of the pendulum to the sum of the masses of
the ball and pendulum. Compare this ratio with the experimental value for
the fractional loss of kinetic energy computed in part (c). Note: This is an
example of a completely inelastic collision (i.e., the colliding objects stick
together after the collision) in which the fractional kinetic energy loss is
maximized.
(e)
Is there a violation of the conservation of energy when the ball collides
with the pendulum? What happens to the missing kinetic energy? Explain.
Explain why momentum can be conserved throughout the entire collision process
in spite of the presence of dissipative (non-conservative) forces between the
pendulum and the steel ball at the time of impact. Dissipative forces, as all forces,
must obey the Newton’s third law.
6 Conclusion
This section should have a clear statement of the results of the experiment and the extent to
which the results are in agreement with the theory being tested. Include percent error.
To make this comparison meaningful, you should include the impact of the experimental error on
your results. Please include a statement of what you have learned, a critique of the experiment,
and any suggestions you have which you think could improve the experiment or the lab handout.
Physics 181 - Summer 2013 - Experiment #5
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The Conservation of Momentum in One Dimension
Data Sheet
1. Record the masses of the steel ball and the pendulum. Numbers to two places after decimal
point.
mball = __________ grams
Mpendulum = __________ grams
Estimated error in reading balance. Error = ± m = ± M = _______ grams
2. Measurement of hi and hf of the Pendulum. Numbers to two places after decimal point.
#
1
2
3
4
hi
(cm)
5
#
1
2
3
4
5
#
6
7
8
9
10
hf
(cm)
hf
(cm)
Estimated error in reading yellow plastic ruler. Error = ± hi = ± hf = _______ cm
3. Measurement of the vertical height y and the range X of the steel ball. Numbers to two places
after decimal point.
#
1
2
3
4
5
y
(cm)
#
1
2
3
4
5
X
(cm)
#
6
7
8
9
10
X
(cm)
Estimated error in reading 2 meter metric ruler. Error = ± y = ± X1 = _______ cm
Estimated error in reading yellow plastic ruler. Error = ± X2 = _______ cm
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4 Calculations
4.1.1. Use a calculator to obtain the mean value of h . Keep three numbers after the decimal
point to reduce rounding-off errors in subsequent calculations. Enter correct units.
h = hf - h i
hi = ____________ Standard Deviation s = ____________ SDM = ____________
hf = ____________ Standard Deviation s = ____________ SDM = ____________
h = hf - hi =____________
4.1.2. Use a calculator to obtain the mean value of y . Keep three numbers after the decimal
point to reduce rounding-off errors in subsequent calculations. Enter correct units.
y = ____________ Standard Deviation s = ____________ SDM = ____________
4.1.3. Use a calculator to obtain the mean value of X . Keep three numbers after the decimal
point to reduce rounding-off errors in subsequent calculations. Enter correct units.
X = ____________ Standard Deviation s = ____________ SDM = ____________
4.2.
Calculate the systems initial momentum.
pi = m X
g
2y
pi = ____________ g-cm/s
4.3.
(Example calculation 4.425x104 g-cm/s)
Calculate the final momentum of the system just after the collision.
p f = (M + m) 2g h
pf = ____________ g-cm/s
(Example calculation 4.375x104 g-cm/s)