Lab 126 Conservation of Momentum and Impulse Momentum Theorem

1. Lab 126:Conservation of Momentum and Impulse – Momentum Theorem
2. Introduction

For a body of mass moving with velocity v, the linear momentum p is defined in the equation p = mv, where momentum and velocity act in the same direction. Momentum is conserved provided there is no net external force acting on the system. For this reason, collisions between elements of the same system don’t change the total linear momentum.

For an external force acting during time interval Δt, the impulse J is defined in the equation J = F Δt. According to Newton’s Laws of Motion and as stated before, changing the motion of an object requires an external force as seen in the Impulse-Momentum Theorem, J = Δp = p_f - p_i. According to this theorem, the amount of momentum change of an object in time interval t equals the net impulse of the same time period.

Three types of collisions exist, fully elastic, inelastic and totally inelastic. In elastic collisions, objects bounce off of each other and kinetic energy and momentum are conserved. In inelastic collisions, objects don’t bounce, and momentum is conserved while kinetic energy is not. In special cases of inelastic collisions, objects stick together in totally inelastic collisions so their final velocities match.

In any collision, as long as a system is considered isolated, the total momentum of a system is conserved. In this lab, we measured glider velocities before and after collisions in both elastic and inelastic situations on a frictionless air track. The conservation of momentum and kinetic energy was confirmed for various cases. In addition, the Impulse-Momentum Theorem was observed in Part III, where change in momentum of a glider was used to verify the relationships expressed by this theory.

3. Experimental Procedure

Part 1. Conservation of Momentum in Elastic Collision

1. In this part of the experiment, the air track was set up with two photogates, 1 after the first glider (M1), then another after the second glider (M2). M1 is heavier, while M2 is lighter. An elastic collision was simulated using rubber band yokes attached to the gliders.
2. After setting up the gliders with attachments, the mass of each glider with attachments was recorded in Data Table 1.
3. The two gliders were placed on the air track. The air supply was adjusted for minimal friction.
4. The air track was leveled.
5. The “Lab 126” file was opened to the “Part 1: Elastic Collision” tab.
6. The photogates were plugged into the 850 Universal Interface and the digital port.
7. The length of the reflector was measured using the caliper and was input as the “ReflectorLength” in the “Calculator” box on the left of the software screen.
8. The appropriate amount of masses for each trial was placed on M2.
9. The record button was clicked. M1 was gently pushed towards M2.
10. For each trial, data from the program was recorded in the data table.
11. When all the velocity measurements were done, the “STOP” button at the bottom of the screen was pressed and photogates were disconnected.
12. For each of the 3 collisions, momentum and KE before and after collisions were calculated. Momentum and KE were verified to be matching and conserved.

Part 2. Conservation of Momentum in Inelastic Collision

1. In this part of the lab, the two gliders, M1 and M2, were outfitted with a needle tip and a wax receptacle so they would stick together to simulate a perfectly inelastic collision. A motion sensor was placed at one end of the air track and connected to the interface.
2. The mass of each glider with attachments was measured, and values recorded in Data Table II.
3. The “Part II” page on the software screen was selected.
4. The “RECORD” button was clicked to start the experiment. M1 was pushed gently towards M2. Distance v. time and velocity v. time graphs were generated in the program.
5. The velocities before and after collision of both of the gliders was observed with the curve fitting and active data tool. The initial and final velocities of both gliders were obtained and recorded in the data table.
6. A few trials were performed with increasing masses on the second glider M2. Data was recorded in the data table.
7. The KE and momentum before and after collision in the system was calculated. That momentum was conserved while KE was not was verified to be true of inelastic collisions.

Part 3. Impulse-Momentum Theorem

1. The “Part III” page of the software was clicked.
2. For this trial, only one glider was present in the experiment. This glider was outfitted with a rubber band yoke, just like the force probe located in a spot for collision. A motion detector was placed at the end of the track. This setup was created to test the Impulse-Momentum Theorem.
3. The motion sensor and force sensor were connected to “PasPort 1” and “PasPort 2”, respectively. The force sensor was zeroed.
4. The reflector and bumper were attached to the glider and the total mass of the glider was measured and recorded in Data Table III.
5. The glider was placed close to the motion sensor on the air track.
6. The “RECORD” button was pressed to start the experiment. The glider was pushed towards the force sensor, rendered immobile. Distance v. time and force v. time graphs were generated in the program.
7. The impulse of FΔt or impulse was determined by using the area tool. Initial velocity, final velocity, and impulse were recorded in Data Table III.
8. The experiment was repeated with several differing glider masses.
9. Momentum before and after collision were calculated. The impulse was compared with the change in momentum to verify the Impulse-Momentum Theorem.
10. Results (Data and Conclusion)

4.1 Experimental Data

Part 1 momentum and KE

Table 1

Part 2 momentum and KE

Table 2

Part 3 momentum

Table 3

4.2 Calculations

The data in the charts were given, while momentum and KE were calculated.

Calculations of change in momentum and kinetic energy are as followed:

Since under conservation of momentum or conservation of KE, p_i should equal p_f and KE_i should equal KE_f, the following equations and their relationships would hold true:

KE:

KE_i = 1/2m_1iv_1i² + 1/2m_2iv_2i²

KE_f = 1/2m_1fv_1f² + 1/2m_2fv_2f²

KE_i = KE_f under conservation of energy

Momentum:

p_i = m_1iv_1i + m_2iv_2i

p_f = m_1fv_1f + m_2fv_2f

p_i = p_f under conservation of momentum

Using the relationships shown in these equations, I calculated initial momentum, final momentum, initial KE and final KE. Sample calculations for momentum and kinetic energy are as followed: (Part 1, trial 1)

KE_i = 1/2m_1iv_1i² + 1/2m_2iv_2i² 

KE_i = ½(0.1983 kg)(0.4813 m/s)² +½(0.1983 kg)(0 m/s)²

KE_i = ½(0.1983)(0.4813)² +½(0.1983)(0)²

KE_i = ½(0.1983)(0.2316) + 0

KE_i = 0.023 J

KE_f = 1/2m_1fv_1f² + 1/2m_2fv_2f²

KE_f = ½(0.1983 kg)(0 m/s)² +½(0.1983 kg)(0.5335 m/s)²

KE_f = ½(0.1983)(0)² +½(0.1983)(0.5335)²

KE_f = 0 +½(0.1983)(0.2846)

KE_f = 0.028 J

p_i = m_1iv_1i + m_2iv_2i

p_i = (0.1983 kg)(0.4813 m/s) + (0.1983 kg)(0 m/s)

p_i = (0.1983)(0.4813) + (0.1983)(0)

p_i = (0.1983)(0.4813) + 0

p_i = 0.095 kg x m/s

p_f = m_1fv_1f + m_2fv_2f

p_f = (0.1983 kg)(0 m/s) + (0.1983 kg)(0.5335 m/s)

p_f = (0.1983)(0) + (0.1983)(0.5335)

p_f = 0 + (0.1983)(0.5335)

p_f = 0.106 kg x m/s

4.3 Error Analysis

There was significant error in this experiment. As seen in the data, the results for KE and PE were not always desired or predictable. In addition to that, the execution of different trials likely varied widely because there was no real specifications on the force applied to the different trials when pushing the gliders. Likely the results were a bit impacted either lower or higher depending. Sample error calculations and the prepared error tables are as seen below:

Part I error table:

Sample error calculation: (for part 1 trial 1)

Error was calculated the same way for all 3 parts of the lab.

Theoretical/True Value:

Energy Computed: 0.028 J

Experimental Value:

Energy Measured: 0.023 J

Trial 1 error

E_abs = observed value – true value

E_abs = 0.023 J – 0.028 J = -0.005 J

0.023 J ± 0.005 J

E_rel = E_abs /true value

E_rel = 0.005 J /0.028 J

E_rel = 17.9%

For Part II, the results were supposed to confirm the conservation of momentum and the lack of conservation of KE. Percentage difference between initial and final momentums and kinetic energy calculations are as below.

The average percent difference in the trials for momentum was 32.4%, while the average percent difference for the KE trials was 99%. It somewhat verifiable, then, that with these results, one can conclude momentum is conserved in inelastic collisions while the same cannot be said to be true of KE.

For Part III of the experiment, the impulse J seen on the program was compared to the change in momentum to find a percent difference to verify the Impulse-Momentum Theorem, which states that Δp = J. The error chart is as seen below.

5. Discussion

In a comparison of momentum and kinetic energy values for all the trials in all 3 parts of the lab, the relationships spoken through the laws of conservation of energy and momentum were roughly verified. Percent error ranged from around 3% - 150%. This degree of error was likely mostly due to not conducting the trials carefully enough, pushing the glider gently and properly setting up equipment in the right place for maximum accuracy. It is unlikely the program recorded data incorrectly because some trials were very close to expected results, while other trials were completely off. The simplest explanation for this degree of error is human error in conducting trials and preparing conditions of the experiment itself. Even small changes in setup, like placing the motion detector too high, or placing the photogates slightly too far apart, could have had drastic effects on attained data and configured results. The error was significant, and to prevent this same problem next time and improve upon these results, I would suggest carefully performing trials and taking care that the equipment setup is good for optimal results.

Besides that, sources of error could have included errors of reading the program velocities and other data and mass from the scale used to mass equipment, error in manually recording mass, velocity, and other variable measurements, failing to perform an adequate number of trials as there was only 1 trial for every changed KE or momentum system, error in setting up the experiment, for instance, placing too many or too few masses on gliders, failing to completely level the air track before performing the experiment, failing to zero the force sensor, forgetting to place the appropriate flag or reflector on top of gliders, affixing photogates at angles from the vertical, placing rubber band yokes at clashing angles, among other things. Systematic error could have occurred because supposing the equipment setup, which was very important and changed for every lab part, was slightly off for each part of the lab, which is highly likely, or there was some other inconsistency not accounted for, results could have been slightly skewed up or down. Photogates could have been placed at the wrong places, at angles, the track could have been tilted, the force sensor could have been positioned too low or high. The air machine to prevent friction could have been turned on too high, where the gliders slid when they shouldn’t have, or too low, where friction present would mean calculations of energy and momentum would not be conserved due to loss of energy in transit. Random errors could have occurred as well. Different people performing different trials meant that characteristic ways of spinning the apparatus could have affected the results to be a little lower or higher. It could have also occurred that the orientation of the equipment slightly changed each trial due to previous trials or unaccounted for experiences. It was notable that the rubber band yokes were a bit loose in the glider slots. Due to this design flaw, it is highly likely that each trial meant the orientation of the rubber band yokes changed a little and the force detected each trial was randomly affected as a result. Other causes of random error could have included malfunctioning equipment and its related considerations.

In Part I, the percentage of momentum and kinetic energy lost during the collisions could be determined through the error rates. For reference, please look back to the error section above to see the percentage of momentum and kinetic energy found lost. The possible sources of energy and momentum loss were friction, air pressure, or even accidental transfer of energy through an applied force on the system. Conservation of momentum and kinetic energy require that there are no external forces acting on the system. Otherwise, work and impulse must be considered as well for all energy and momentum to be accounted for. The “lost” energy likely transferred into other types of energy, or internal energy manifestations, like sound and heat.

The fractional KE loss is theoretically supposed to be calculated with the expression M2/M1 + M2. Using this expression, I calculated that the fractional energy loss (converted to percentages) would be, in order for the 3 trials of Part II, 48.7%, 59.2%, and 66.1%. The actual fractional energy loss are 21.2% and 30.3% and 69.3% in the same order. Looking at this data, I can say that the relationship is confirmed with experimental results. Granted, there is a considerable degree of error, but as this was explained earlier, some error is to be expected.

6. Conclusion

As this lab is concluded, I now understand the concept of conservation of momentum along with elastic and inelastic collisions. I also know how to verify the Impulse-Momentum Theorem. The basic idea is that a system without any external forces will have conservation of momentum. In the case an external force is applied, the system has impulse. Objects that face collision may participate in one of two types, either elastic or inelastic collisions. Elastic collisions occur when objects bounce off of each other. Under conditions for an elastic collision where no external forces are applied, KE and momentum are conserved. Under conditions for inelastic collision where no external forces are applied, momentum is conserved while KE is not. This lab may not have been performed as ideally as liked, but I understand the possible error factors involved as listed in the error section, and were the experiment performed again, I would make a greater effort a second time around to perform consistent trials, taking care not to extend more force than needed as not to unintentionally add impulse to the system.

For this lab, we completed 3 parts of the experiment as listed in the lab booklet. Part I dealt with conservation of momentum in elastic collisions. Part II dealt with conservation of momentum in inelastic collisions, and part III dealt with the Impulse-Momentum Theorem. For each part of the lab, various masses of gliders were placed on a frictionless air track to try to simulate conditions for differing collisions. A program was then run and values of velocity and impulse (for Part III) were recorded in data tables. From the data tables, conservation of momentum and KE in certain cases were confirmed. The percent differences between expected results based on theory and actual results were then compiled and placed into error graphs.