1:
2:
(ta initials)
first name (print)
last name (print)
brock id (ab13cd)
(lab date)
Experiment 3
Collisions and conservation laws
In this Experiment you will learn
• about the principle of the conservation of momentum
• the law of conservation of energy applied to Q, a ration that measures energy loss during an interaction
• to make a collision record of two colliding pucks
• to analyse the quality of vector dot data and properly draw vectors on a sheet of paper
• to correctly perform a graphical vector addition, with careful transposition and measurement of
vectors
• that you can apply a divide-and-conquer self-checking strategy to complex expressions, so that the
chances of algebraic or calculation errors are greatly reduced.
Prelab preparation
Print a copy of this Experiment to bring to your scheduled lab session. The data, observations and
notes entered on these pages will be needed when you write your lab report and as reference material
during your final exam. Compile these printouts to create a lab book for the course.
To perform this Experiment and the Webwork Prelab Test successfully you need to be familiar with the
content of this document and that of the following FLAP modules (www.physics.brocku.ca/PPLATO).
Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the
module in depth, then try the exit test. Check off the box when a module is completed.
FLAP PHYS 1-2: Errors and uncertainty
FLAP MATH 2-1: Introducing geometry
FLAP MATH 2-4: Introducing scalars and vectors
Webwork: the Prelab Colliding Pucks Test must be completed before the lab session
Important! Bring a printout of your Webwork test results and your lab schedule for review by the
!
TAs before the lab session begins. You will not be allowed to perform this Experiment unless the
required Webwork module has been completed and you are scheduled to perform the lab on that day.
Important! Be sure to have every page of this printout signed by a TA before you leave at the end
!
of the lab session. All your work needs to be kept for review by the instructor, if so requested.
CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT!
18
19
Conservation of momentum
The velocity ~v of a body of mass m is determined by its speed, a scalar quantity, and its direction of motion,
represented by a unit vector. If we consider a collision of two such masses m1 and m2 with velocities ~v1b and
~v2b before the collision, and velocities ~v1a and ~v2a after the collision, we note that it is generally difficult,
if not impossible, to predict these resulting velocities ~v1a and ~v2a . To accomplish this, one would need to
have a complete knowledge of the physical characteristics of the objects (size, shape, et cetera) and of the
geometry of the interaction.
The linear momentum P~ of a body is a vector equal to the product of its mass m (a scalar) and its
velocity ~v (a vector). The law of conservation of linear momentum states that:
If there are no net external forces acting on the masses, then the total momentum P~b before the
collision is equal to the total momentum P~a after the collision.
A mathematical formulation of this law for a collision between two masses m1 and m2 is a vector equation
and can be expressed as follows:
P~1b + P~2b = P~1a + P~2a
m1~v1b + m2~v2b = m1~v1a + m2~v2a .
(3.1)
Rearranging Equation 3.1 in terms of the change in the momentum ∆P~1 of m1 and ∆P~2 of m2 reveals
that the net change will be zero and that the vectors will be oriented anti-parallel to one another:
P~1b − P~1a + P~2b − P~2a = 0 → (P~1b − P~1a ) = −(P~2b − P~2a ) → ∆P~1 = −∆P~2
Similarly, the change in the velocity for the two masses will result in two anti-parallel vectors:
m1 (~v1b − ~v1a ) = −m2 (~v2b − ~v2a ) → m1 ∆~v1 = −m2 ∆~v2
(3.2)
Since the velocity vectors are collinear, the vector equation can be simplified to a scalar equation and
expressed in terms of the magnitudes of the velocities |~v2a − ~v2b | and |~v1b − ~v1a |. Rearranging Equation 3.2
as a ratio of masses m1 and m2 :
m1
|∆~v2 |
|~v2b − ~v2a |
|~v2a − ~v2b |
=−
=−
=
.
m2
|∆~v1 |
|~v1b − ~v1a |
|~v1b − ~v1a |
(3.3)
The mass ratio equation predicts that for equal masses m1 and m2 , the change in the velocity of the
two masses should be the same. It also predicts that the two resulting velocity vectors will point in the
same direction since the the components of vector 1 have been reversed.
It is also of interest to know whether kinetic energy K = mv 2 /2 is conserved during a collision. The
parameter Q, defined as the ratio of the total kinetic energy Ka after the collision to the total kinetic
energy Kb before, is used as a measure of the energy lost during a collision. Note that Q depends on the
square of the velocity and hence will be very sensitive to variations in v.
Q=
Ka
=
Kb
1
2
2 m1 v1a
1
2
2 m1 v1b
2
2 + m v2
+ 12 m2 v2a
m1 v1a
2 2a
=
2 + m v2
2
m1 v1b
+ 12 m2 v2b
2 2b
(3.4)
The kinetic energy, and therefore Q, are scalar quantities. Q can range in value from 0 to 1. If Q = 1,
the kinetic energy of the system is conserved and the collision is said to be elastic. A collision is said to
be inelastic when Q < 1. This is the expected result of our experiment since some of the kinetic energy
is changed to heat and sound energy during the collision, and there will be frictional forces acting on the
masses throughout the interaction. On the other hand, rotational kinetic energy may be imparted onto
the pucks as they are pushed.
20
EXPERIMENT 3. COLLISIONS AND CONSERVATION LAWS
Procedure
In this experiment the mass ratio of two colliding pucks will be calculated to determine whether or not
linear momentum was conserved during the collision. The ratio Q will be used to estimate the kinetic
energy lost during the interaction.
The equipment consists of a flat glass plate in a metal
frame which can be levelled by varying the height of four
adjustable legs. The glass plate is covered by a conductive black pad.
A sheet of regular white paper is placed over this
conducting layer. The sheet must be flat and free of any
kinks or other deposits.
The collision components are two heavy metal pucks.
Each puck is tethered to a plastic hose that provides the
puck with compressed air and carries within it an electrode wire. The hose should hang freely, without any
twists or interference from the rest of the equipment.
Figure 3.1: Trails left by moving pucks
When turned on, the air exits from a hole at the bottom of the puck, creating a high pressure layer
between the puck and the paper, causing the puck to levitate and move with negligible friction. When the
frame is properly levelled, the puck will float in place, without any lateral movement.
A spark timer is used to provide the high voltage pulses required in the experiment. One terminal of
the timer is connected through the electrode wire in the hose to an insulated needle at the bottom of the
puck.
The other terminal of the spark timer is connected to the conductive sheet, completing the electric
circuit. Sparks are generated from the needle through the white paper to the conducting paper beneath.
These pulses produce black spots on the bottom of the white paper, marking the positions of the pucks at
equal time intervals ”u”. Puck speeds would then be expressed in units of cm/u or mm/u.
The firing rate in units of u−1 of the timer can be adjusted with a control knob on the unit. The timer
will produce sparks when the pushbutton switch at the front of the unit is pressed. To make a collision
record:
1. Place a new sheet of white paper over the conducting layer, making sure that the paper is flat.
2. Very slowly turn on the air until the pucks begin to float. Excessive air pressure will burst the air
hose. Level the frame by adjusting the height of the four legs until the pucks float in place.
3. Place the pucks against the launchers in adjacent corners of the frame, and release them toward the
centre of the paper, where they will collide. Do several trial runs. While any collision is theoretically
valid, for ease of analysis your collision should be fairly symmetric as in Figure 3.1 and take about
2-3 seconds to complete.
Small initial velocities will cause the pucks to slow down due to friction and yield incorrect results.
The pucks should not rotate as this would give them rotational kinetic energy. We are concerned
only with translational kinetic energy.
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4. To record the collision, instruct an assistant to press and hold the switch of the spark timer just
before you release the pucks, and to release it when the pucks rebound from the edges of the frame.
Every member of the team needs to make their own collision record to analyse.
5. The mass of each puck has been measured and is displayed on the puck. Assign this mass value to
the corresponding trace for future reference, keeping in mind that the traces appear on the bottom
of the paper. Be sure to match each mass with the corresponding trace, otherwise your Q value
will not make sense.
6. A good collision record will show for each of the four trails a series of 4-6 equadistant and collinear
dots, spanning a length greater than 100 mm. The length is measured with a ruler scale graduated
in millimetres. If your collision did not turn out, flip the sheet of paper paper and try again.
? Is the suggested minimum length of 100 mm significant? Does the measurement error depend on the
length of the a vector? If so, how would you minimize the measurement error?
Data analysis
It can be seen from Equation 3.3 that the generation of the vectors ~v1b , ~v2b , ~v1a , and ~v2a is required.
For example, the vector ~v1b can be obtained from
the collision record by joining a series of dots along
the Puck 1 trail (before the collision) spanning n
time intervals.
• Draw four proper vectors on your sheet,
(line segment begins and ends at a dot, arrowhead ends at a dot) as shown in Figure 3.2.
It is important to use this same number n of
time intervals for all the vectors so that you
are, in effect, applying the same time scale
n ∗ u to each vector.
Figure 3.2: Join dots to obtain vectors
• Measure the distance between the adjacent dots of each trail and enter the results and observations
in Table 3.
vector
|v1b |
|v1a |
|v2b |
|v2a |
yes/no
yes/no
yes/no
yes/no
number of segments in vector
dot spacing first segment (mm)
dot spacing last segment (mm)
vector length (mm)
dots are collinear?
Table 3.1: Experimental data: vector analysis results
? Evaluate the quality of your collision record. Did the distance between the dots in each trail remain
the same and did these dots form a straight line? Is the trail a valid representation of a vector?
22
EXPERIMENT 3. COLLISIONS AND CONSERVATION LAWS
m1
|v1b |
|v1a |
|~v1b − ~v1a |
Time units
m2
|v2b |
|v2a |
|~v2a − ~v2b |
(g)
(mm)
(mm)
(mm)
(u)
(g)
(mm)
(mm)
(mm)
Table 3.2: Experimental data: masses and vector magnitudes
The vectors (~v2a − ~v2b ) and (~v1b − ~v1a ) depend
on the magnitude and direction of the component vectors, hence a graphical vector addition
must be performed. A reasonably long vector or line
segment with well defined endpoints can be transposed very accurately by visually estimating the final placement of this vector as shown in Figure 3.3:
• Using a long ruler, extend the line segment
AB beyond the region of point D.
• Place the ruler so that the edge rests on point
C and is parallel to the previously drawn line.
• Draw a line through C beyond points A and
D.
• Measure near A and D the perpendicular
distance between the two lines to verify that
they are indeed parallel.
Figure 3.3: Transposition of a vector
• Determine the length AB with a ruler and draw a line segment of length AB from C to D to define
the vector CD. Enter your results in Table 3.
When your vector transpositions are completed, have a TA review your collision record and vector
!
addition. Make sure to have the TA sign and date your collision record. You need to keep this
experimental data as part of your lab notes for possible use at a later date.
By Equation 3.2, the change in momentum of one puck is equal and opposite to that of the other puck.
The two resultant vectors should then be nearly equal in magnitude since m1 ≈ m2 but parallel since the
a − b components in the vectors have been reversed in Equation 3.3.
? Compare the lengths of vectors (~v2a − ~v2b ) and (~v1b − ~v1a ). Does the result make sense in light of the
preceding explanation?
• To quantify the test for parallelism, extend the vector lengths as above and calculate the angle θ, in
degrees, between the two extended line segments. It is given by θ = arctan(dy/dx), where dy = y2 −y1
is the difference in the separations y1 and y2 at the opposite ends of the two lines of length dx.
dx = .............
y1 = .............
y2 = .............
dy = ..................... = ..................... = .....................
θ = ..................... = ..................... = .....................
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• Determine a theoretical value (m1 /m2 ) and error ∆m1 /m2 for the mass ratio from the given values
of m1 and m2 . The measurement error for these masses is ±0.1 g.
∆
m1
m2
= ..................... = ..................... = .....................
= ..................... = ..................... = .....................
m1
m2
(Theoretical)
m1
= ............... ± ...............
m2
• Use Equation 3.3 to calculate a value and error for the experimental mass ratio of the pucks.
∆
m1
m2
= ..................... = ..................... = .....................
= ..................... = ..................... = .....................
m1
m2
(Experimental)
m1
= ............... ± ...............
m2
• Use Equation 3.4 and the theoretical m1 and m2 values determine Q and ∆Q.
When solving equations such as for Q and ∆Q, a good technique is to split the equation into several
terms and solve for each term separately. This avoids repetition and will expose possible calculation errors.
For example, there are four identical mv 2 terms, each of which appears several times in the Q and ∆Q
equations. Evaluate each term only once, then compare the four results. You would expect them to be
similar since the m and v values are also similar.
Likewise, the equations for each error term are identical in form and should also yield similar results.
You can then confidently complete the calculation of Q and ∆Q.
2
A = m1 v1a
= ..................... = .....................
2
B = m2 v2a
= ..................... = .....................
2
C = m1 v1b
= ..................... = .....................
2
D = m2 v2b
= ..................... = .....................
A+B
C +D
= ..................... = .....................
Q=
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EXPERIMENT 3. COLLISIONS AND CONSERVATION LAWS
s
∆m1
m1
2
s
∆m2
m2
2
∆C = C
s
∆m1
m1
2
∆D = D
s
∆m2
m2
2
2
= ..................... = .....................
2
= ..................... = .....................
2
= ..................... = .....................
2
= ..................... = .....................
(∆A)2 + (∆B)2 (∆C)2 + (∆D)2
+
(A + B)2
(C + D)2
= ..................... = .....................
∆A = A
∆B = B
∆Q = Q
s
∆v1a
+ 2
v1a
∆v2a
+ 2
v2a
∆v1b
+ 2
v1b
∆v2b
+ 2
v2b
Q = ............... ± ...............
Note: Keep your collision record for your own further analysis or for review by the TA or professor, if
requested.
IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK!
Lab report
Go to your course homepage on Sakai (Resources, Lab templates) to access the online lab report worksheet
for this experiment. The worksheet has to be completed as instructed and sent to Turnitin before the lab
report submission deadline, at 11:00pm six days following your scheduled lab session. Turnitin will not
accept submissions after the due date. Unsubmitted lab reports are assigned a grade of zero.
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