Collisions in One Dimension Paul Erlanger November 12, 2017 Date Performed: October 30, 2017 Partner: Alex Song TA: Chris Tiede Abstract In this lab we sought to investigate the conservation of energy and momentum by exploring collisions in one dimension using two different bodies. We explored both elastic and inelastic collisions of these bodies using different mechanisms do allow for maximum/ minimal friction. We also investigated how the conservation laws would change (or if they would change at all) if we used different massed bodies. 1 Elastic Collisions 1.1 Objective To compare velocities of two different bodies before and after an elastic collision to explore the conservation laws hold up. To explore how changes in mass and friction further change our calculations. 1.2 Materials: Gliders Air track Photogate Sensors Capstone Software Index Cards Magnets 1.3 Measurements & Results We first measured the length and mass of all three carts and got the following data. Cart 1: length = 13.1 +/-.05 cm, mass= 299 +/-.5 grams, Cart 2: length = 13.0 +/-.05 cm, mass= 299 +/-.5 grams, Cart 3: length = 13.0 +/-.05 cm, mass= 448 +/- 1 gram We completed multiple trials using many different orientations of elastic collisions, the following data was collected: 1
Using this data we collected both the initial and final momentum and Kinetic Energy for each trial and got the following data: 2
NOTE: IN LAB NOTEBOOK TRIALS WITH M1 > M2 SHOULD BE LABELED TRIALS M2 > M1 AND VICE VERSA. We know that the % change in P and Ke should both be 0. For ease of analysis I created two columns showing the percent difference in both momentum and kinetic energy for all trials. We were relatively close with our analysis. Our smallest difference was in Trial 8 with a.63 % change in momentum and a 12% change in Kinetic energy. All of our trials were pretty accurate and likely within the error of the experiments. Note that our errors were generally smaller when using magnets because of a reduction in kinetic energy loss and that kinetic energy change was on balance greater than the difference in momentum. This is because any error in velocity is increased when squared to find kinetic energy. 1.4 Analysis & Response Questions 1. The incident glider changes direction when the still glider has a higher mass. This makes sense because all of the energy is transferred to making the other car move and then some. The negative velocity is necessary to balance out momentum and kinetic energy. Similarly when you throw a ball against the wall it will reverse direction because the wall is much more massive. 2. To make the best possible measurement we decided to place the sensors as close as possible where they will be able to read both the cards. This will make sure that error is minimized and that we will get the best possible reading right after the collision. Further, in the experiments where both carts moved with a positive final velocity, one cart needed to be stopped so the second sensor would accuratly read the second cart, this has no effect on measurements because the stoped cart had previously been read. 3
3. During our lab we estimated the uncertainty of measured quantities to be the following based on our believed accuracy of the measurements. Mass of car 1 and 2 uncertainty=.5 grams, mass of car 3 uncertainty = 1 gram, all length uncertainties (of cards) were.05 cm. The only uncertainty in velocity would come from the length in the card because the photo-gate sensor measures the time exact to many decimal places. So uncertainty in velocity (if measured in cm/s) is.05. 4. To check if the conservation is within our measured results within the error propagated we must first propagate the error. Using trial ones initial velocity we can check if we were within the propagated error for momentum and energy. Error when mass of the cars is the same: k = 1/2mv 2, δv = p = mv, δp = ( δp δm δm)2 + ( δp δv δv)2 = ±14.951 ( δk δm δm)2 + ( δk δv δv)2 = (1/2 (.29) 2.5) 2 + (299.29.05) 2 = ±4.34 Based on Run 1, we can conclude that both our change in kinetic energy and momentum s were within our expected errors. Our change in momentum was 9 and our change in kinetic energy was 2.5. Both of these are within our errors because both quantities should be 0 in an elastic collision. 5. Yes our results are sensical considering that energy is loss to heat when contact occurs. In a perfectly elastic collision, theoretically there should be no change in kinetic energy but since our system is not idealized and there are places that energy is loss, even though we tried to minimize them, it makes sense that in none of our trials ke was lost and the change in ke was greater (percent wise) then our change in momentum on average. 6. Kinetic energy is loss to heat when the two cars collide and also whatever minimal friction/ resistance is given to the cars, only in an idealized world will there actually be a perfect conservation of energy. As for trials of magnet vs. the cars bumping, we actually saw better conservation in trials without the magnets. Theoretically, this should not be the case because there is less friction when the come into contact with magnets because the cars never really touch, however I may be mis-informed about this. Assuming that is true, the reason our trials came out differently is because our trials with the magnets were not very accurate because the magnets were very finicky. They often would shift and even collide causing a loss in energy. 1.5 Conclusion I can conclude that we successfully were able to show that momentum and energy is conserved in an elastic collision within a reasonable degree of certainty. Further, we were able to discuss where energy is lost in the system because our system is not idealized perfectly and also reason why our trials with magnets were less successful than those without. 4
2 Perfectly Inelastic Collision 2.1 Objective To compare velocities of two different bodies before and after a perfectly inelastic collision to explore the conservation rules under these conditions. To explore how changes in masses and different holding mechanisms further change our calculations. 2.2 Materials: Gliders Air track Photogate Sensors Capstone Software Index Cards Wax mechanism Velcrow 2.3 Measurements & Results We used the same carts with the same lengths and masses to complete the following trials: First we used two cars with the same mass to complete 25 trials using a wax mechanism to make the cars stick together. Therefore cart 1 moved at an initial velocity, game into contact with cart 2 and remained stuck with wax traveling at a single velocity as a single system. Here is the data and analysis from these trials (sorry the trials are out of order): 5
We then used this data to find initial, final momentum and energy and the percent change of both: 6
This data is sensible, momentum should be conserved as you can see by all our trials, however because energy is lost (maximum energy) in an inelastic collision we lost 50% of our kinetic energy in each trial. This makes sense because the move at half the speed after the trial because they are the same mass and the mass effectively doubles with the same velocity after the collision, this reveals that momentum is conserved even when energy is not. We then conducted the same 25 trials but using Velcro instead, again with same massed cars, we got the following data: And here is our analysis: 7
Once again sorry for the mix up in runs but the columns are consistent for both data and analysis. There was one obvious outlier (Run 4 ) this was just an error in capstone reading. On average there was more fluctuation in the Velcro readings because it was a less stable mechanism to make them stick (the Velcro often would move and unstick to the cars.) However you can see that our change in momentum which is the second to last column was near 0 on average and our change in kinetic energy was about 50 % on average, the same as the wax trials. There was one more set of data we were asked to collect and that is measurements of fluctuations in mass, that data can be seen here: 8
Some things to note are that more kinetic energy is lost when the cart that is initially at rest is the more massive one and that clearly the trials with wax had less error because the percent differences in momentum were significantly smaller. 2.4 Analysis & Response Questions 1. To check if the conservation is within our measured results within the error propagated we will use the same method as we did in the elastic trials (looking at a specific data point). Using trial one s initial velocity, from the trial with the cars with the same mass we will check if we were within our error. Note since this is an inelastic collision with cars of the same mass, we expect there to be a 50% decrease in kinetic energy but no change in momentum. Error when mass of the cars is the same: p = mv, δp = ( δp δm δm)2 + ( δp δv δv)2 = ±14.952 Note that there is very little change in the uncertainty from the one measured above, this makes sense because the only thing that changed is the velocity and ever so slightly k = 1/2mv 2, δv = ( δk δm δm)2 + ( δk δv δv)2 = (1/2 (.47) 2.5) 2 + (299.47.05) 2 = ±7.03 Our change in momentum was 14.95 which is just inside our error range, meaning that the change in momentum was expected if you take into account maximum error. Our change in kinetic energy was 20 while you would expect a 16.5 Joule loss. This is also within our expected error of 7.03 joules. 9
2. Yes in all trials above there can be seen an average of 50% kinetic energy loss when the carts were the same mass, a 30-40% loss when the smaller mass was initially at rest and about a 60% loss when the cart with the larger mass is originally at rest. The kinetic energy goes into other forms of energy like heat and is lost when the carts stick together. 3. The wax trials were a lot more successful in terms of minimizing error than the Velcro experiments. This was because there were still movement in the car in the Velcro trials while in the wax trials the cars instantly stuck together. Further the Velcro was a lot more finicky than the wax and this can be verified by almost a perfect conservation of momentum in the wax trials but not the same conservation in the trials with the velcro. 4. Since the lab did not specify which of the 25 trials we had to use we decided to go with our 25 trials using wax for the following calculations because we felt they would be more accurate. Here is a picture of the python code, the output and the produced graph: 10
2.5 Conclusion I can conclude that we were able to successfully evaluate change in energy and momentum under various types of inelastic collisions. We fluctuated mass and realized when the more massive car is the one at rest initially, more energy was lost then when this car was the one moving. This makes sense because more energy is used to get the larger car moving when in rest. We were also able to determine that our measurements for when the cars were the same mass were within our error to the expected change in kinetic energy (50%) and our expected change in momentum (0). 11