The Conservation Of Momentum Environmental Sciences Essay

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 Î”P = Pf-Pi =-8.595 and Î”KE = KEf-Kei = -4.762. In an inelastic collision of equal massess Î”P = -12.989 and Î”KE = -43.14. In an explosive collision of equal massess Î”P = -448.038 and Î”KE = -118.211.

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 = Â½ 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:

Î”KESYS = KEsys,final – KEsys,intial

For the two collisions stated earlier if Î”KESYS is equal to zero the collision is considered elastic, however if Î”KESYS 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

0.483

251.65

244.4946

62.9125

59.04545

0.494

0.482

248.6302

243.9884

61.41166

58.8012

0.574

0.505

288.8942

255.631

82.91264

64.54683

0.422

0.405

212.3926

205.011

44.81484

41.51473

Î”P = Pf-Pi

0.482

0.496

242.5906

251.0752

58.46433

62.26665

-8.595433333

0.516

0.498

259.7028

252.0876

67.00332

62.76981

Î”KE = 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

0.501

294.3059

237.5554

84.94838

63.52835

0.502

0.59

310.6885

245.6916

126.1154

88.10411

0.321

0.466

324.3081

244.3456

51.56687

54.96218

0.462

0.544

337.2292

242.4102

106.818

74.9014

Î”P = Pf-Pi

0.51

0.602

354.5463

242.5007

130.167

91.72445

-81.71491849

0.486

0.52

324.2156

242.5007

118.2043

68.43824

Î”KE = 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

0.309

468.8014

310.1742

79.76525

47.92191

0.396

0.243

495.1158

243.9234

39.46275

29.63669

0.697

0.351

523.2297

352.3338

122.2538

61.83458

0.554

0.296

563.0325

297.1248

77.23541

43.97447

Î”P = Pf-Pi

0.596

0.343

610.7959

344.3034

89.39011

59.04803

-227.7090311

0.493

0.278

532.195

279.0564

61.16328

38.78884

Î”KE = 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

311.5427

291.7455

96.42247

42.15722

0.602

0.276

0.274

302.9866

277.6096

91.19897

38.17143

Î”P = 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

Î”KE = 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

352.9982

470.2152

124.6126

73.35357

Î”P = 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

Î”KE = 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

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

573.035

280.3206

79.76525

26.06982

Î”P = 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

Î”KE = 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

0.482

0.503

497.2092

122.4709

0.448

0.471

463.8986

106.6245

0.489

0.512

505.2881

126.4901

0.438

0.469

457.8532

103.9089

Î”P = Pf-Pi

0.478

0.492

489.6278

118.7447

488.0378833

0.506

0.513

514.3504

131.0292

Î”KE = KEf-KEi

average

488.0379

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

0.297

0.615

608.5803

139.8729

0.34

0.618

653.1376

154.517

0.292

0.619

605.6006

139.6484

0.307

0.633

627.7009

148.5813

Î”P = Pf-Pi

0.276

0.574

566.8072

121.5127

599.3574667

0.24

0.581

534.3182

114.2626

Î”KE = KEf-KEi

average

599.3575

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.

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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