The conservation of momentum was shown in three types of collisions, elastic, inelastic and explosive. By getting mass and velocities for two carts during the collision the change in momentum and kinetic energy was found. In an elastic collision of equal massess ΔP = Pf-Pi =-8.595 and ΔKE = KEf-Kei = -4.762. In an inelastic collision of equal massess ΔP = -12.989 and ΔKE = -43.14. In an explosive collision of equal massess ΔP = -448.038 and ΔKE = -118.211.
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This shows that conservation of momentum is conserved in elastic and inelastic equations due to their very low change in momentum; however kinetic energy is conserved in the elastic collision but not in the inelastic collision. In an explosive collision momentum is not conserved since the two objects start at rest with no momentum and gain momentum once moving opposite.
Introduction
Just like Newton’s laws, the conservation of momentum is a fundamental principal in physics that is integral in daily life. However unlike Newton’s laws, the conservation of momentum does not seem to be entirely intuitive. If a ball is thrown in the air some momentum seems to be loss to the air. This makes proving the conservation of momentum tricky and difficult to do in a real life setting.
To measure the conservation of momentum in the lab, two carts will be used along a frictionless track. This allows calculation to be easier since the vectors will be moving along only one axis. This way positive direction can be movement to the right while negative direction can be movement to the left. One cart will have a plunger which is ejected by a spring that will convert its potential energy to kinetic energy of the cart. This will knock the other cart and its momentum will be transferred either partially or entirely. These velocities of the two carts will be measured by a graphing device. This is shown in diagram 1.
Diagram 1.
Momentum is produced by mass and velocity, in other words:
p = mv.
It is important to point out that momentum is not conserved on an object by object basis, however it is conserved for the isolated system. This is shown in the equation:
Psystem = P1 + P2.
Therefore if momentum is conserved then the initial momentum of the entire system should equal the final momentum of the entire system. Thus this can be shown in the equation where:
Psystem, initial = Psystem, final
M1 X V1i + M2 X V2i = M1 X V1f + M2 X V2f
In the lab collisions will be shown to illustrate the conservation of momentum. In elastic collisions energy is always conserved. Unfortunately for this lab kinetic energy can be converted into heat so that energy is lost to viable measurements. If the energy is conserved, the collision is considered to be elastic, but if the energy is not conserved, then the collision is considered inelastic.
Kinetic energy is energy associated with motion where an object with mass and moving with a certain velocity the equation is:
KE = ½ m |v|2
This allows to find the loss or gain in energy of a system much like for momentum where the change in kinetic energy of a system is determined by the equation:
ΔKESYS = KEsys,final – KEsys,intial
For the two collisions stated earlier if ΔKESYS is equal to zero the collision is considered elastic, however if ΔKESYS does not equal zero then the collision is considered inelastic. There is also another type of collision that will be determined in this lab called an explosive collision. This can be considered the opposite of an inelastic collision since the energy is not conserved because the kinetic energy is transformed for potential energy to kinetic energy. These three types of collisions will be measured in the lab under differing conditions and the change in momentum and kinetic energy of the system will be calculated.
Procedure
In the lab the momentum and kinetic energy will be calculated by measuring different velocities for the two carts at different masses. Two carts will be set along a frictionless track. As stated earlier this allows for easier calculations since it allows working only in one dimension. One of the carts used has a plunger while the other car is just a regular car. Both carts have different sides which will allow the emulation of the different collision types.
For and elastic collision the plunger cart will be placed against the side of the ramp and then set off by a small piece of wood. It will the knock the other cart and emulate a elastic collision because the carts have magnets facing each other that will help conserve energy and momentum by having the opposite sides face each other. Having magnets of opposite charge face each other help keep the collision elastic since major contact between the two carts can convert kinetic energy into heat and will be lost. This will be done in three different ways, first having equal mass carts, second having the plunger cart heavier than the regular cart, and lastly by having the plunger cart lighter than the regular cart. The velocities for these carts will be measured for the different variable for six different trails and averaged.
For the inelastic the set up will be identical except to emulate this collision the carts will have Velcro sides that will be facing each other and cause the carts to stick together once they hit each other. This will be done in three different ways similar to the elastic collision, first having equal mass carts, second having the plunger cart heavier than the regular cart, and lastly by having the plunger cart lighter than the regular cart. The velocities for these carts will be measured for the different variable for six different trails and averaged also.
For the explosive collision the two carts will be sitting next to each other. The plunger car will have its plunger faced toward the adjacent regular car so when the button is pressed the will move away from each other in opposite directions. This will only be done in two different ways, one way having the carts equal in mass and one ways have one cart heavier than the other cart. The velocities for these carts will be measured for the different variable for six different trails and averaged as well.
Results
Table 1. Elastic Collision Data
Elastic –
Equal Mass
regular car (g) –
506.2
plunger car (g) –
503.3
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
Kef= .5m1vf1 + .v5m2vf2
0.5
0
0.483
251.65
244.4946
62.9125
59.04545
0.494
0
0.482
248.6302
243.9884
61.41166
58.8012
0.574
0
0.505
288.8942
255.631
82.91264
64.54683
0.422
0
0.405
212.3926
205.011
44.81484
41.51473
ΔP = Pf-Pi
0.482
0
0.496
242.5906
251.0752
58.46433
62.26665
-8.595433333
0.516
0
0.498
259.7028
252.0876
67.00332
62.76981
ΔKE = KEf-KEi
average
250.6434
242.048
62.91988
58.15744
-4.762437183
Elastic –
Heavy Int.
regular car (g) –
506.2
plunger car (g) –
1000.9
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
Kef= .5m1vf1 + .v5m2vf2
0.412
0
0.501
294.3059
237.5554
84.94838
63.52835
0.502
0
0.59
310.6885
245.6916
126.1154
88.10411
0.321
0
0.466
324.3081
244.3456
51.56687
54.96218
0.462
0
0.544
337.2292
242.4102
106.818
74.9014
ΔP = Pf-Pi
0.51
0
0.602
354.5463
242.5007
130.167
91.72445
-81.71491849
0.486
0
0.52
324.2156
242.5007
118.2043
68.43824
ΔKE = KEf-KEi
average
324.2156
242.5007
102.97
73.60979
-29.36021623
Elastic –
Light Int.
regular car (g) –
1003.8
plunger car (g) –
503.3
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
Kef= .5m1vf1 + .v5m2vf2
0.563
0
0.309
468.8014
310.1742
79.76525
47.92191
0.396
0
0.243
495.1158
243.9234
39.46275
29.63669
0.697
0
0.351
523.2297
352.3338
122.2538
61.83458
0.554
0
0.296
563.0325
297.1248
77.23541
43.97447
ΔP = Pf-Pi
0.596
0
0.343
610.7959
344.3034
89.39011
59.04803
-227.7090311
0.493
0
0.278
532.195
279.0564
61.16328
38.78884
ΔKE = KEf-KEi
average
532.195
304.486
78.21177
46.86742
-31.34434946
For the elastic collision with equal masses the change in momentum and kinetic energy is every small. Where as in the other two methods the change in momentum is much larger since the masses where different then the change in kinetic energy.
Table 2. Inelastic Collision Data
Inelastic –
Equal Mass
regular car (g) –
506.2
plunger car (g) –
503.3
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
Kef= .5m1vf1 + .v5m2vf2
0.622
0.292
0.297
313.0526
297.305
97.35936
43.78238
0.481
0.242
0.243
242.0873
244.8052
58.222
29.68293
0.619
0.289
0.289
311.5427
291.7455
96.42247
42.15722
0.602
0.276
0.274
302.9866
277.6096
91.19897
38.17143
ΔP = Pf-Pi
0.51
0.236
0.237
256.683
238.7482
65.45417
28.23227
-12.98885
0.502
0.248
0.249
252.6566
250.8622
63.41681
31.16993
ΔKE = KEf-KEi
average
279.8348
266.846
78.67896
35.5327
-43.14626406
Inelastic –
Heavy Int.
regular car (g) –
506.2
plunger car (g) –
1000.9
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
0.495
0.322
0.321
319.6722
484.78
122.6228
77.96833
0.506
0.343
0.342
323.0093
516.4291
128.1332
88.48103
0.497
0.317
0.318
336.2746
478.2569
123.6157
75.8842
0.499
0.312
0.312
352.9982
470.2152
124.6126
73.35357
ΔP = Pf-Pi
0.323
0.211
0.208
367.6309
316.4795
52.21145
33.23065
115.4745216
0.486
0.31
0.308
339.917
466.1886
118.2043
72.10332
ΔKE = KEf-KEi
average
339.917
455.3916
111.5667
70.17019
-41.39646683
Inelastic –
Light Int.
regular car (g) –
1003.8
plunger car (g) –
503.3
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
0.575
0.181
0.181
480.8526
272.7851
83.20178
24.68705
0.589
0.172
0.163
506.4235
250.187
87.30267
20.77979
0.555
0.179
0.183
534.182
273.7861
77.51449
24.87125
0.563
0.186
0.186
573.035
280.3206
79.76525
26.06982
ΔP = Pf-Pi
0.367
0.115
0.113
619.6586
171.3089
33.89449
9.736832
-289.887818
0.574
0.178
0.179
542.8304
269.2676
82.91264
24.05466
ΔKE = KEf-KEi
average
542.8304
252.9426
74.09855
21.6999
-52.3986526
For the inelastic collision the change in kinetic energy is much larger then it was in elastic collision. This holds true for the other all three methods used.
Table 3. Explosive Collision Data
Explosive –
Equal
regular car (g) –
506.2
plunger car (g) –
503.3
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
Kef= .5m1vf1 + .v5m2vf2
0
0.482
0.503
0
497.2092
0
122.4709
0
0.448
0.471
0
463.8986
0
106.6245
0
0.489
0.512
0
505.2881
0
126.4901
0
0.438
0.469
0
457.8532
0
103.9089
ΔP = Pf-Pi
0
0.478
0.492
0
489.6278
0
118.7447
488.0378833
0
0.506
0.513
0
514.3504
0
131.0292
ΔKE = KEf-KEi
average
0
488.0379
0
118.2114
118.2113751
Explosive-
Unequal
regular car (g) –
506.2
plunger car (g) –
1000.9
v1 (m/2)
v1f (m/s)
v2f (m/s)
Pi = m1vi1+ m2 vi2
Pf = m1vf1 + m2 vf2
Kei = .5m1vi1 + .v5m2vi2
Kef= .5m1vf1 + .v5m2vf2
0
0.297
0.615
0
608.5803
0
139.8729
0
0.34
0.618
0
653.1376
0
154.517
0
0.292
0.619
0
605.6006
0
139.6484
0
0.307
0.633
0
627.7009
0
148.5813
ΔP = Pf-Pi
0
0.276
0.574
0
566.8072
0
121.5127
599.3574667
0
0.24
0.581
0
534.3182
0
114.2626
ΔKE = KEf-KEi
average
0
599.3575
0
136.3992
136.399151
For the explosive collision the change in momentum is much larger than in the other two collisions. There is no initial momentum for this collision since the two carts started together at rest.
Conclusion
From momentum and the kinetic energies calculated from the formulas the different trails were averaged to find the initial and final momentum and kinetic energy for each of the eight conditions. They the change in momentum of the system was calculated for the system by subtracting the final momentum minus the initial momentum. This was then done for kinetic energy to find the change in kinetic energy by subtracting final minus initial as well. This produced different values for the different conditions.
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For the elastic collision the momentum and kinetic energy are supposed to be conserved. As table 1 shows, the momentum and kinetic energy for the equal mass carts is very close to zero, much closer than for the other conditions. For the heavier plunger cart, the initial force had much more inertia and caused the lighter second car to move much further. This is opposite in the other conditions where the plunger cart was much light. It had a harder time moving the second heavier cart. The main difference for the change in momentum and kinetic energy for the two unequal mass cart conditions was due to the fact the final velocity for cart one was never measured properly. It was assumed that the velocity was zero when in fact the plunger cart moved slightly after the collision. The assumption was due to careless human error.
For the inelastic collision kinetic energy is not conserved. This is evident very much in the results for the change in kinetic energy. There is a much larger value or this change then in the elastic counterpart since the carts stick together and move as one unit. This close interaction allows for the loss of energy as heat. As for the explosive collision, the change in momentum is by far the largest. Since the system start at rest it is entirely potential energy. When the collision happened the carts move apart and become kinetic energy. Since the final momentum is subtracted by an initial momentum of zero, it is obvious why the change is so large.
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