Download Collisions - faculty at Chemeketa

Collisions
When two objects collide, they exert forces on one another. If we know the exact forces and
time of contact, then we can use Newton’s second law and kinematics equations to predict
the subsequent motion of the objects. But the exact forces exerted can be extremely difficult
to measure, and they often vary over the course of the collision, so a different method of
analysis is preferred. The preferred analysis uses the law of conservation of momentum.
Momentum can be calculated with the following definition:
[1]
p = mv
p = momentum
m = mass
v = velocity
Newton’s second law can be rewritten in terms of momentum as follows:
[2]
[3]
[4]
[5]
ΣF = ma
ΣF = m(Δv/Δt)
ΣF = Δ(mv)/Δt
ΣF = Δp/Δt
(Newton’s second law)
(definition of acceleration)
(assumption of constant mass)
(definition of momentum)
The second law can be stated as “the net force is the rate of change in momentum.”
If we analyze a pair of objects with no significant net external force, then we can use
equation [5] and Newton’s third law to prove conservation of momentum.
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
F21 = Δp1/Δt
(equation [5] applied to object 1)
F12 = Δp2/Δt
(equation [5] applied to object 2)
F21 = -F12
(Newton’s third law)
F21 + F12 = 0
Δp1/Δt + Δp2/Δt = 0 (equations [6] and [7] in equation [9])
Δp1 + Δp2 = 0
(multiply by Δt)
(p1f - p1i) + (p2f - p2i) = 0 (definition of Δ)
p1i + p2i = p1f + p2f
For a system with no significant net external force, the total momentum before an
event equals the total momentum after an event. This is called “conservation of
momentum.” This analysis can be extended to more than two objects. In this lab, we will
test the motion of two objects in one dimension. Conservation of momentum in one
dimension is written as follows:
[14]
[15]
p1ix + p2ix = p1fx + p2fx
m1v1ix + m2v2ix = m1v1fx + m2v2fx
If two objects collide without significant transformations of energy to sound, thermal
energy, or permanent deformation, then kinetic energy will be conserved. This special type
of collision is called “elastic.”
[16]
[17]
K1i + K2i = K1f + K2f
½m1v1ix2 + ½m2v2ix2 = ½m1v1fx2 + ½m2v2fx2
If the collision is not elastic, then the expectation is that the total kinetic energy after the
collision is less than the total kinetic energy before the collision.
If two objects stick together after a collision, then the collision is called “completely
inelastic.” The conservation of momentum equation can be rewritten for this special case as
follows:
[18]
[19]
m1v1i + m2v2i = m1vf + m2vf
m1v1i + m2v2i = (m1 + m2)vf
Physics is fun!
Experimental Procedures
Error Estimation for Velocity
1) Obtain a single glider, measure its length, and place it at one end of the air track.
2) Place one photogate near the glider and place a second photogate near the end of the
air track.
3) Turn on the air track and photogates.
4) Send the glider through the photogates.
5) Observe the time measurement on each photogate.
6) Calculate the velocity of the glider at each photogate using distance/time.
7) Calculate the difference in the velocity. This value can be used as the uncertainty in
subsequent velocity calculations.
Collisions
8) Obtain two gliders.
9) Measure the mass and length of each glider.
10) Place the gliders on the air track and turn on the air track. One possible
configuration is illustrated above. If you choose to have one of the gliders be a
stationary target, it will need to be placed between the photogates.
11) Configure two timers to the correct height so that they measure the time it takes a
glider to pass. The timers should be set to “gate” and “memory on”.
12) Shove one or both of the gliders so that they collide. If they collide with their Velcro
sides, then it will be an inelastic collision. If the gliders collide with their bumper
sides, then it might be an elastic collision.
13) Immediately record a description of the collision. For example, you might say
something like, “A yellow glider was initially stationary in the middle of the track. We
shoved a red glider to the right. It collided with the yellow glider and bounced off,
moving to the left. The yellow glider moved to the right after the collision.
14) Take time readings from the timers. The display on the timer will be the time it took
a glider to pass the first time. If a glider passed a photogate more than once, then flip
the memory switch to “read” and it will give you the total elapsed time. You will
need to subtract to obtain the time for a glider to pass the second time.
15) Calculate (or infer a zero if stationary) the velocity of each glider before and after the
collision. You can use distance over time for magnitude of the velocity because it is
not accelerating. You will need to decide on a coordinate system and indicate
positive or negative for each velocity. Immediately record this information. If
you do not do this, you might get a zero on your report and be required to
repeat the experiment.
16) Use equation [15] to calculate if momentum was conserved in the collision. Be sure
to use positive or negative velocity as needed in your calculations to indicate
direction. The expectation is that momentum will be conserved in any collision.
Note: you may use equation [19] instead if this is an inelastic collision.
17) Use equation [17] to calculate if kinetic energy was conserved in the collision. The
expectation is that kinetic energy will be conserved (the same total initial and final)
for an elastic collision and transformed (initial greater than final) for any other type
of collision.
18) Repeat the experiment using different initial velocities, different type of collision, or
different mass of glider. Complete a total of four trials.
Note: You will have four calculations of total initial momentum, four calculations of total
final momentum, four comparisons of total initial momentum to total final
momentum, four calculations of total initial kinetic energy, four calculations of total final
kinetic energy, and four comparisons of total initial kinetic energy to total final kinetic
energy. Because you are doing highly repetitive calculations, it is recommended that you use
a spreadsheet.
Note on units: You do not need
to worry about exclusively using
SI units in this experiment as
long as you consistently use the
same system of units. You are
welcome to use CGS
(centimeter, gram, second) units,
for example. The CGS unit of
momentum is the g*cm/s. The
CGS unit of energy is the “erg” which is a g*cm2/s2.