Chapter 6_ Conservation of Momentum in 1D

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Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel Abstract: In this experiment, we used the theory of conservation of momentum to develop a model of conservation of momentum and analyzed the data by comparing final momentum to initial momentum through cart collisions. The results of this experiment indicated that as the initial momentum and velocity increased, so did the final velocity and momentum. The results show that momentum is conserved. Participants in this experiment gained a better understanding of conservation of momentum in 1D while also seeing how an isolated system conserves its total momentum. Introduction: Conservation of momentum says that in any isolated system, the total momentum of that system will be conserved. That the total quantity of momentum in a given isolated system will remain constant. An isolated system is a system in which no external forces are acting upon. This experiment is 1D kinematics meaning that it is constrained to one dimension. A mathematical equation for the conservation of momentum is m 1 v 1i + m 2 v 2i = m 1 v 1f +m 2 v 2f . The law of conservation of momentum is only valid for an isolated system. There are two types of collisions, “elastic” and “inelastic”. An elastic collision is a collision in which a momentum and kinetic energy are conserved. An inelastic collision is a collision in which kinetic energy is not conserved, where the colliding object ends by sticking firmly together. This experiment is testing the law of conservation of momentum by using inelastic collision via cart collision. The apparatus used in this experiment consisted of a dynamic track, track bumpers, two smart carts, cart masses, and a spring cart launcher. Each cart featured a magnetic bumper at one end and corresponding Velcro tabs at the opposite end. In this experiment, the model is representing the initial and final momenta of an inelastic collision between two objects of different masses which is represented by the equation, m 1 v 1i + m 2 v 2i = (m 1 +m 2 ) v f . This equation simplified is m 1 v 1i = Mv f because the initial velocity of v 2i is 0 m/s, m 2 v 2i cancels out. In the simplified equation, m 1 stands for the mass (kg) of cart 1, v 1 signifies the initial velocity (m/s) of cart 1, m 2 stands for the mass (kg) of cart 2, M signifies the total mass (kg) of m 1 and m 2 , and v f represents the final velocity (m/s) of the two carts after collision. This is the equation used because during the collision of the two carts, the Velcro and magnet attachment causes them to adhere, essentially turning them into a single object with the same velocity. Some assumptions that can be made in the model equation is that force of friction on the cart is negligible, the track is level and horizontal, the cart is a sliding frictionless object, the magnitude of g is constant, the carts are constrained to move in one direction, and that the system is isolated. The least reliable assumption is the system is isolated because there can be a possibility of external forces acting upon the cart that may affect the momentum.

Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel

Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel Procedure: Before starting the experiment, we created a model graph to represent the final momentum versus initial momentum by using the equation m 1 v 1i + m 2 v 2i = m 1 v 1f +m 2 v 2. We did this to analyze the conservation of momentum and to test against our experimental results. To begin the experiment we set up PASCO software and connected the smart car to PASCO in order to be able to record the data. Then, we weighed the mass of the cart which was .318kg. Our apparatus consisted of two smart carts, a spring cart launcher, and a track where we placed the carts. The opposite ends of each cart has matching velcro tabs. The springs we used in our lab were compression springs which take force to compress from the starting rest position. Once they are compressed and fully released, it exerts a constant force in the opposite direction causing Cart 1 to move forward and collide with Cart 2 resulting in both carts sticking firmly together by the velcro tabs. We first placed the resting cart (Cart 2) at the 70cm mark on the track (this was kept the same throughout all trials). Then, using the spring cart launcher we pulled back the cart and released it causing an inelastic collision between Cart 1 (the cart with the spring launcher attached) and Cart 2 (the resting cart). We repeated this process for six trials, pulling the spring launcher farther back for each trial. For each trial we collected data using a Position (m) vs. Time (s) graph. By using the Position (m) vs. Time (s) graph we were able to get the initial velocity by looking at the slope of the line, as well as the final velocity which we obtained by looking at the slope of the Position vs Time graph after the collision of Cart 1 and Cart 2. Then, we recorded this data for all six trials which can be seen in Data Table 1. Once we got this data, we were able to calculate the initial momentum and final momentum for each trial by using the equation , where p = momentum, m= mass of the cart (which was weighed before 𝑝 = 𝑚𝑣 starting the experiment, .318kg), and v= velocity. Since we recorded six different trials on the Position (m) vs. Time (s) graph, we were able to calculate the initial momentum and final momentum for all six trials which can be seen in Data Table 2. Lastly, we used Excel to create a plot of Initial Momentum vs. Final momentum based on our experimental data and included both the experimental data and theoretical model in our graphical representation which can be seen in Graph 2. We also used Excel to plot the Final momentum (kg ⋅ m/s) vs. Final velocity (m/s), including both our experimental data and theoretical model in our graph, which is displayed in Graph 3. Data: V i V f Trial 1 0.849 0.204 Trial 2 0.690 0.327 Trial 3 0.746 0.353 Trial 4 0.716 0.335 Trial 5 1.06 0.345

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Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel Trial 6 1.10 0.322 Table 1 - Experimental Values of Initial and Final Velocity P i P f Trial 1 .270 .065 Trial 2 .219 .104 Trial 3 .237 .112 Trial 4 .228 .107 Trial 5 .337 .109 Trial 6 .350 .102 Table 2 - Initial and Final Momentum using M*V Graph 1 - Theoretical Model of the Initial and Final Momenta of a Totally Inelastic Collision Between Two Objects of Different Masses

Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel Graph 2 - Comparison of Model and Experimental Initial and Final Momenta Graph 3 - Final Momentum vs. Final Velocity (m/s) in Model and Experimental Data Mass Cart 1: 0.318kg Mass cart 2: 0.292kg

Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel Analysis: In Table 1, our raw data is shown. We obtained this by recording data changing our velocity 6 different times. The table shows our initial velocity, and then the final velocity after making contact with the other cart. Table 2 is a table showing the Initial and Final momentum of the cart using mass * velocity, using the mass of the cart which was 0.318 kg. Graph 1 was created using model data to find an initial and final momentum to base our experimental data on. Graph 2 was created by plotting our final and initial momentum from our experimental data in Table 2 over our model data. Graph 3 was created by plotting our final momentum and final velocity in our model data and final velocity and final momentum in our experimental data on the same graph to compare them as well. Based on our comparison between final and initial momentum of the model and experiment, we can see that our model describes the trend of our experiment well, but most likely due to error is not exactly the same. They both show a general positive increasing trend, but the model data has more of a steep slope and increases faster than our experimental data. This was due to some outlying points which were most likely due to errors made during the experiment. The relationship between our independent variable and final momentum was a positive relation. From this, you can draw the conclusion that velocity is affected by momentum and they have a positive correlation. The sources of error in our experiment were primarily systematic of possible ways velocity could have been decreased. A significant error that is an example of this was when the cart would drift slightly off track when attaching to the other cart. We would try to be careful about this when pulling the cart back, and record the data again if this did happen. Another source of error would be if the track is level or not which would be a random error and could skew the data either way. This was not able to be tested using our equipment, so if we could add any piece of apparatus, I would include measuring the level of the track. Some aspect that I would change in our procedure is using something better than velcro like tape or something that will make it stick and replace it every time. The velcro didn’t work great sometimes, and maybe the more you use it the less adhesive it gets. Something else in our procedure that could have been adjusted to improve our results would be measuring how far back we pull the spring cart launcher in order to get more accurate results of the velocity. One significant source of error in our apparatus was when we released the cart after pulling the spring launcher back, it would go off the track as it moved forwards. Our suggested improvements/replacement would address this source of error by making sure the track is level so that the cart is stable on the track. We believe this error was a systematic error because the experimental values were all similar and not all over the place. Conclusion: For this experiment, we demonstrated the law of conservation of momentum by calculating initial and final momentum. We did this by creating a model of an inelastic collision. The data shows us that as initial momentum increased so did the final momentum and as the initial velocity increases so did the final velocity. The model was not a good representation of the experiment since it can be seen that the experimental value line was not similar to the model because of an error we had. Even though the experimental values were increasing, it was not a

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Group 4 Chloe Kalahar, Maricella Velita, and Esha Patel significant increase like the model data values. Despite this, we would say we are pretty confident in our results due to the fact that it does agree to the model data. Something we can do that can ensure better results is that when we do our trials, we can make sure our pull of spring is a little more than before since we were not measuring the pull of our spring each time. Also, make sure we don’t have an error like we had before or do an extra trial to get better results.

Chapter 6_ Conservation of Momentum in 1D

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