Conservation of Momentum Lab Report

Conservation of Momentum Introduction The objective of this laboratory experiment is to demonstrate conservation of linear momentum in one-dimensional collisions of objects, and then compare the properties of elastic and inelastic collisions. To achieve this objective, the experiment makes use of two gliders colliding and passing through photogates along a frictionless air track as a means to demonstrate the principle of linear momentum conservation. We will be monitoring the initial and final velocities of the two gliders in three different collision scenarios: the small glider colliding into a stationary big glider, a big glider colliding into a stationary small glider, and a big glider colliding into the small glider. We will also be monitoring the time that it takes each glider to pass through the photogates. By using velocities of each of these cases, along with the time intervals associated with each glider passing through the photogates, we can calculate p and KE for the system and experimentally verify the conservation of linear momentum. The different quantities that are essential in this experiment are w, m, M, v, p and KE. w is the width of the metal tab atop the small and big glider which is what passes through the photogate and is measured in cm. m and M are the masses of the small and big gliders respectively and their values are measured in grams. v represents the velocity at which the gliders are traveling at before and after colliding. p represents the linear momentum of each object before and after the collision, and KE represents the kinetic energy of each object. The relevant equations for this lab are v = w t which represents the velocity of each glider, p = mv which is the linear momentum of each object before and after colliding, and 2 1 KE = 2 mv which is the formula to calculate the kinetic energy of each object. Procedure The equipment used in this laboratory experiment was an air track, a small glider, a big glider, and 2 photogates. Three different collisions were conducted with the two gliders. o An elastic collision: small glider colliding with the bigger glider o An elastic collision: big glider colliding with the smaller glider o An inelastic collision: big glider colliding and sticking with the small glider. Two photogates were placed on the air track and labeled as 1 and 2. For the first collision, we placed the small glider at one end of the air track and the big glider in the middle of the track. They were positioned so that the spring ends of the two gliders were facing each other. o We released the smaller glider, it rebounded off the big glider, and passed through the same photogate it entered, and the big glider passed through the opposite photogate that the smaller glider passed through. o Each glider was removed from the track once it exited through its respective photogate For the second collision, we launched the big glider into the small glider. They were both allowed to pass through their respective photogates and were removed from the air track once they exited through the photogates (same procedure as previous step, just the gliders are switched). For the final collision, there were a few changes. o We positioned the gliders so that their Velcro ends were facing each other. o We recorded the time when the big glider entered through the photogate precollision, and when the small glider connected with the big glider, due to the Velcro, exited through the photogate post-collision. Data m = 118 g = 0.118 kg σm =1 g = 0.001kg wm = 4.9 cm = 0.049m σw = 0.002m M = 219 g = 0.219 kg σM =1 g = 0.001kg wM = 4.9 cm = 0.049m σw = 0.002m Data table 1 Elastic Collision sliding small glider into big glider Glider ti (s) vi σvi vf σvf pi σpi pf σpf (m/s) (m/s) (m/s) (m/s) (kgm/s) (kgm/s) (kgm/s) (kgm/s) tf (s) Small 0.053819 0.209109 0.9105 0.0372 0.082395 Big KEi (J) σKEi (J) KEf (J) σKEf (J) 0.0096 0.1074 0.0045 0.0489 0.0029 0.0032 0.0002 0.0276 0.0012 0.2343 0.5947 0.0243 0.1302 0.0053 0.0387 0.0022 Data table 2 Elastic collision sliding big glider into small glider Glider ti (s) Small Big vi σvi vf σvf pi σpi pf σpf (m/s) (m/s) (m/s) (m/s) (kgm/s) (kgm/s) (kgm/s) (kgm/s) tf (s) 0.055986 0.8752 0.0357 0.228501 0.231289 0.2144 0.0088 0.2119 0.0086 0.0470 0.0019 KEi (J) σKEi (J) KEf (J) σKEf (J) 0.1033 0.0043 0.0452 0.0026 0.0464 0.0001 0.0050 0.0003 0.0049 0.0004 Data table 3 Inelastic collision sliding big glider into small glider with velcro ti (s) Pre 0.081025 Post tf (s) --0.126435 vi (m/s) σvi vf σvf KEi pi σpi pf σpf (m/s) (m/s) (m/s) (kgm/s) (kgm/s) (kgm/s) (kgm/s) (J) 0.1325 0.0055 0.6048 0.0247 0.3876 0.0158 σKEi (J) KEf (J) σKEf (J) 0.0401 0.0023 0.1306 0.0054 0.0253 0.0015 Data Analysis The first thing that my lab partner and I calculated was the initial and final velocity of the w small and big glider for each case. By using the formula v = t , the initial and final velocities could be calculated. For demonstrative purposes, the initial and final velocity for the small glider during case 1 will be shown below. vi = wm ti 0.049 m = 0.053819 s =0.9105 m/ s vf = wm tf 0.049 m =−0. 2343 m/s = 0.209109 s Since velocity is a vector quantity, its measurement must take into account not only the magnitude but the direction of motion as well. Since the glider began to travel in the opposite direction post collision, the final velocity must reflect this direction change which is why its value is negative. By substituting our values for each case into the equation, we were able to obtain the initial velocities of 0.9105 m/s (for the small glider in case 1), 0.2144m/s (for the big glider in case 2), and 0.6048m/s (for the pre-collision in case 3). Since the big glider was stationary in case one, and the small glider was stationary in case 2, they did not have initial velocity values. This was the same case for the final velocities, where by substituting our values into the equation we were able to obtain final velocity values of -0.2343m/s (for the small glider in case 1), 0.5947m/s (for the big glider in case 1), 0.8752 m/s (small glider case 2), 0.2119m/s (big glider case 2), and 0.3876m/s (post collision case 3). The reason why the final velocity of the big glider in case 2 was not negative was because during the experiment, after the big glider had collided with the stationary small glider it did not change direction (as was the case in the first collision with the small glider). Since the big glider did not change direction and continued to travel in its original direction post collision, the final velocity did not need to be negative. Next, the uncertainty for the initial and final velocity values were calculated. This was done by using the equation σv = σw t . An example of this calculation is demonstrated below using the small glider’s values for case 1. σvi = σwm ti = 0.002 m =¿ 0.053819 s 0.0372 m/s By substituting our values for each case and for each glider into this equation we were able to obtain the uncertainty values for both the initial and final velocities that are seen in data tables 1, 2, and 3. The initial and final momentum values for each glider during each case were then calculated. By using the formula p = mv, the initial and final momentum values can be calculated. An example of this calculation can be seen below where the initial and final momentum of the small glider in case 1 is calculated. ( ) pi=mm v i=0.118 kg 0.9105 m =0.1074 kgm/ s s ( ) pf =mm v f =0.118 kg −0.2343 m =−0.0276 kgm/s s Since momentum, just like velocity, is a vector quantity, its measurement must consider the magnitude and direction of motion of the object. Since the smaller glider began to travel in the opposite direction after colliding with the big glider, its momentum must be negative. By substituting our other initial and final velocity values into this equation, we were able to obtain the initial and final momentum values see in data tables 1-3 for the small and big glider respective to their velocity values. Next, we calculated the uncertainty for the initial and final momentum values in each case. This propagation of the uncertainties was calculated using the equation σv ¿ 2 v σm ¿2 +¿ . By substituting our values into this equation, we were able to obtain the m ¿ σp = p √ ¿ uncertainty values seen in each data table. A demonstration of this calculation using the small glider during case 1 is shown below. σ vi 2 ¿ vi σm 2 ¿ +¿ = m ¿ σ pi= pi √ ¿ 0.0372 m/ s 2 ¿ 0.9105 m / s ¿ 0.001 kg 2 0.0045 kg*m/s ¿ +¿ 0.118 kg ¿ 0.1074 kg∗m/ s √ ¿ Next, the initial and final kinetic energies for each case were then calculated. Using the 1 mv 2 , we were able to obtain all the initial and final kinetic energy values seen formula KE = 2 in each data table. The values for the small glider during case 1 will be used to demonstrate this calculation. 1 1 KEi = 2 m vi 2 = 2 (0.118kg)( 0.9105 m/s)2 = 0.0489 J 1 1 KEf = 2 m vf 2 = 2 (0.118 kg)(-0.2343 m/s)2 = 0.0032 J Since Kinetic Energy is a scalar quantity, its measurement relies on magnitude solely. This is why the final kinetic energy of the small glider for case 1 was positive even though both its final momentum and final velocity were negative. By substituting our values for each case into the equation, we were able to determine the initial and final Kinetic Energy values seen in the data tables. Finally, my lab partner and I calculated the uncertainty to our initial and final Kinetic 2∗σv 2 ¿ v σm 2 ¿ +¿ . An example of this Energy values. This was done by using the formula, m ¿ σKE= KE √ ¿ calculation is done below for the smaller glider in case 1. 2∗0.0372 m/ s 2 2∗σv i 2 ¿ ¿ 0.9105 m/ s vi ¿ σm 2 0.001 kg ¿ 2+¿ 0.0029 J ¿ +¿ = m 0.118 kg ¿ ¿ σK Ei=K Ei √ ¿ 0.0489 J √ ¿ By substituting our values for each case into this equation we were able to determine the uncertainty values for the initial and final kinetic energies of the big and small glider during each collision. By comparing the momentum values and the overlap of their estimate ranges with the uncertainties for the big and small glider, we can verify if the conservation of momentum was present for each of the three collisions. After analyzing the data to determine overlap, it was determined that the inelastic collision had the greatest overlap out of the 3 collisions, followed by the first elastic collision performed, and the second elastic collision performed had no overlap at all. Therefore, momentum was conserved in the first and third collision, but not the second collision. As for the conservation of kinetic energy, none of the collision cases were able to conserve kinetic energy. Theoretically, the elastic collisions should have conserved Kinetic energy while the inelastic collision did not. When comparing how close the values were to overlapping, the first collision (elastic) was the closest, followed by the third collision (inelastic), and the second collision (elastic) had the greatest difference. Conclusion In this laboratory experiment, we were able to demonstrate the conservation of momentum to an extent. Given the three different collisions, the first being an elastic collision where a 0.118kg glider collides with a stationary 0.219kg glider, the second being an elastic collision where a 0.219kg glider collides with a stationary 0.118 glider, and the third being an inelastic collision where a 0.219kg glider collides and sticks with a 0.118kg glider, the conservation of momentum and kinetic energy for each case was able to be determined. Theoretically, given an isolated system, momentum should be conserved for all three collisions. Although the first and third collision had overlap in their momentum values, thus their momentum was conserved, this was not the case with the second collision where there was no overlap what so ever in its momentum values. There were also issues with the conservation of Kinetic Energy for these collisions. Theoretically, Kinetic Energy should be conserved for elastic collisions in an isolated system whereas Kinetic Energy is not conserved for inelastic collisions because after the objects collide and stick together, the kinetic energy is converted into another form of energy (heat, etc.). Although the lack of overlap with the kinetic energy values for the third collision supported the theoretical expectation since it was inelastic, the lack of overlap for the first and second did not since both of these collisions were considered elastic. These discrepancies with the data may have stemmed from a few sources of error. One of the assumptions made in this experiment is that the surface was level. If the surface is not level, then the glider that was supposed to remain stationary for each collision would not have been able to do so, thus the supposed “stationary” glider would have been moving and therefore its initial velocity would not have been zero which was also assumed. If the initial velocity for this glider was not zero, this would have skewed the results for the momentum and kinetic energy of the system. Another assumption was that the system (air track and gliders) were frictionless. The friction caused by the air track and the air resistance that the gliders were subjected to as they moved along the air track, would have reduced the velocity at which the gliders were traveling with as they passed through the second photogate which records the glider’s final velocity. This decrease in the final velocity values would have altered the final momentum and final kinetic energy calculations, which ultimately would affect the extent to which the momentum and kinetic energy values would overlap which dictates whether momentum and kinetic energy is conserved for these collisions. In order to reduce these discrepancies in this experiment, more trials should be conducted, the surface of the track should be checked prior to the start of the lab to make sure that it is as level as possible, and it should be performed in a vacuum to reduce any friction caused by air resistance.