Wednesday 22 February 2012
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PHYS 116 SCALE-UP
19
2-D collisions
9-11
Rear-Ended
2-D collisions (notebook)
Ponderable: Rear-ended
Mini-lab: 2-D collisions
o Equipment: 1 spark table and 2 air pucks, operated by lecturer
o At least 1 elastic and 1 inelastic 2-D spark track for each group
o Rulers or meter sticks for measurement.
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Wednesday 22 February 2012
PHYS 116 SCALE-UP
REAR-ENDED
To earn extra money while at Carolina, you form an accident-reconstruction team with your classmates.
Your first job is to analyze an accident for the Chapel Hill Police Department. A careless driver rearended a car that was halted at a stop sign. Just before impact, the driver slammed on his brakes, locking
the wheels. The driver of the struck car had his foot solidly on the brake pedal, locking his brakes. The
mass of the struck car was 900 kg, and that of the initially moving vehicle was 1200 kg. On collision, the
bumpers of the two cars meshed. You go out to the crash site and determine from the skid marks that
after the collision the two cars moved 0.76 m together. You also run a few experiments and conclude
that the coefficient of kinetic friction between the tires and pavement was 0.92 on the day of the
accident. The driver of the moving car claims that he was traveling at less than 15 km/hr as he
approached the intersection. Is he telling the truth?
2
Wednesday 22 February 2012
PHYS 116 SCALE-UP
2-D COLLISIONS
Background. Consider two objects with masses m1 and m2 that undergo a glancing collision, one in
which neither mass rebounds in a direction opposite to its initial path. To visualize such a
collision, see the figure to the lower right. Somewhere near the actual collision, a line can be
drawn in an arbitrary direction from which all the angles are measured. Call this line the x-axis.
The angle that the initial (before the collision) path of mass 1 makes with the x-axis is labeled θ1i
and the angle that its final (after the collision) path makes is θ1f. Similar notations are used for the
angles of the initial and final paths of mass 2.
According to the principle of conservation of
momentum, the momentum of the system will
not change as a result of the collision. To
illustrate this idea, suppose the two objects in
the above figure rebound off one another and,
subsequent to the collision, move apart. When
the momenta of each of the two objects have
changed, we say that the objects have exchanged
momentum with each other.
To analyze the collision in the figure, split up the
initial momentum vectors into their components
along the x-axis and the y-axis:
Px,i  P1x,i  P2 x,i  m1v1i cos1i  m2 v2i cos 2i
Py,i  P1y,i  P2 y,i  m1v1i sin 1i  m2 v2i sin  2i
Do the same for the final momentum vectors, keeping in mind the velocity is taken as positive if
the object is moving to the right (in the direction you arbitrarily assign as the positive x
direction) or the object is moving upwards (in the direction that you arbitrarily assign as the
positive y-axis). The sign of the velocity will indicate the sign of each term in the expression for
momentum.
Conservation of momentum requires that Px,i = Px,f and Py,i = Py,f and therefore also that the
total momentum vector, Ptotal , defined by applying the rules of vector addition, is conserved as
well:
Ptotal,i = Ptotal, f
Ptotal = Px2 + Py2
tanqtotal =
3
Py
Px
Wednesday 22 February 2012
PHYS 116 SCALE-UP
If no net work is done during the collision, the collision is said to be elastic, and the kinetic energy is
conserved. For the above example as an elastic collision, the kinetic energy must satisfy:
KEi 


1
1
m1v1i2  m2 v2i2  m1v12f  m2 v22 f  KE f

2
2
If kinetic energy is lost during the collision (i.e.
net work is done on the system), the collision is
said to be inelastic. A diagram of one possible
inelastic case, in which the two objects stick
together after the collision (in which case the
collision is said to be totally inelastic) is shown to
the right:
It is important to note that momentum is
conserved in both elastic and inelastic collisions.
Prelab Deliverable (to be submitted on Sakai before
class):
For equal masses colliding, fill out the missing parts of this table:
initial
final
P1
mv
P2
mv
P1x
P1y
P2x
P2y
-√3mv/2
mv/2
mv/√2
mv/√2
Does it describe an elastic or an inelastic collision?
4
θ1
45°
θ2
30°
Wednesday 22 February 2012
PHYS 116 SCALE-UP
Exploration. For this exercise you will be provided with records of collisions that took place
between pucks on an air table (the air cushion minimizes extraneous effects such as friction
that can interfere with the momentum conservation measurement). As you will see in the
demonstration preceding the lab exercise, a visible record of the motion of the pucks will be
made by creating a spark from the puck to the air table through a piece of recording paper and
carbon paper which have been set on top of the air table.
You will be given sheets of recording paper from this apparatus, containing records of one or
more two-dimensional collisions. Disks moving across the paper will have left marks at equal
time intervals, so that you are able to determine the speed and trajectory of each disk before
and after they collide.
Find the initial speed of each puck just before the collision by measuring the distance between
successive dots near the collision area and dividing by the spark time interval. As you do this,
examine the spacing of the dots. If the spacing is not uniform, what can you conclude about the
speed and the assumptions made for this experiment? Define a set of x and y axes on the paper
and compute your best estimate of the initial velocity (magnitude and direction) for each puck,
along with an estimate of the uncertainty. Record these values in your lab notebook, and use
them to calculate the initial momentum and initial kinetic energy for each puck. From these
values, determine the initial momentum and kinetic energy of the two-puck system. What is
the velocity of the center-of-mass of the two-puck system immediately before the collision?
Find the final velocity, momentum and kinetic energy for each puck in a similar fashion. Make
your velocity measurements as close in time to the collision as possible (consider why this is
important). Use those values to determine the final momentum and kinetic energy of the twopuck system. What is the velocity of the center of mass of the two-puck system immediately
after the collision? Was the collision elastic or inelastic? If kinetic energy is not conserved, can
you account for the missing kinetic energy?
All of your measurements and calculations (as well as the solutions to the applications below)
should be recorded in your laboratory notebooks.
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Wednesday 22 February 2012
PHYS 116 SCALE-UP
Applications
1.
An alpha particle (helium nucleus) and an oxygen nucleus (both initially in motion) collide headon. The alpha particle is scattered at an angle θ1 = 64.0° relative to its initial direction of travel.
The oxygen nucleus recoils at an angle θ2 = 51.0° with a speed 1.20 x 105 m/s. What are the
initial and final speeds of the alpha particle?
2. Two particles P (0.10 kg) and Q (0.30 kg) are released from rest 1.0 m apart. P and Q attract
each other with a constant force of 1.0 x 102 N, and no external forces act on the two particles.
When the separation of the two particles is half of its initial value, what is the speed of the
center of mass of the two-particle system? At what distance from the initial position of P do the
two particles collide?
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