Group3_2_ch6

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**The Law of Conservation of Energy** Andrew, John, Amanda

**Objective:** Find the relationship between changes in kinetic energy and changes in gravitational potential energy.

**Hypothesis:** The initial energy should equal the final energy of the system due to the Law of Conservation of Energy.

**Methods & Materials:**

Station 1: We set the ramp on an incline with a photogate timer at the end. We dropped the cart with the picket fence down the ramp and recorded the time in gate. We measured the change in height and the length of the picket fence, as well as weighed the cart and the fence.



Station 2: We shot the ball out of the horizontal launcher at short range. We measure the diameter of the ball, the initial height, the time in gate, and the weight of the ball.



Station 3: At this station, we used a pendulum. We raise the cork to a height of 34 centimeters (20 centimeters from the bottom of the photogate timer) and dropped it. We measured the time in the photogate, the width of the cork, and the weight.



Station 4: We dropped the metal ball from the short end of the ramp. We measured the weight and diameter of the ball and the height of the ball at its highest point.



Station 5: We launched the ball at short range. We measured the width and weight of the ball and the final height of its path.



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Station 6: We place the ball at the top of the roller coaster ramp. We let the ball drop and measure the time in gate. We also measured the weight of the ball, its width, and the height at the top of the loop.

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<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Our Data Tables:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Class Data Tables:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Sample Calculations:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Analysis:**

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Station 5:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Station 6: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Conclusion:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Our hypothesis that the initial energy and final energy at each station was supported by our data, though not entirely. Because of the Law of Conservation of Energy, we know that the energy at the beginning and end of each station should be the same, and therefore the percent difference should be equal to 0%. Our percent differences, for Stations 1-6, were 4.23%, 26.59%, 6.45%, 13.45%, 14.72%, and 30.30% respectively. Thus, there was a significant amount of error in each lab, especially those that exceeded a 10% difference between initial and final. One source of error for these experiments is friction, which is a force doing work on a moving object, between the surface of that object and another object. We ignored work in these experiments, and therefore also ignored friction, which is usually used to find initial and final energy. Therefore, if we were to find friction and include it in our calculations, we would lower our percent difference as the two energy measurements would be more accurate and thus more similar. Friction would be the largest concern for Station 1, where the cart was moving down an incline. For Stations 2 and 5, where projectiles were used and thus no friction, the high percent differences (26.59% and 14.72%) could be from other sources of error such as the shooters being off by a degree or moving slightly, or in station 2, the ball passing through the second photogate on a slant. To decrease this error, we could ensure that the projectiles were not moved at all, and also use a different instrument for measuring the final velocity that wouldn't vary if the ball came in on a slant. For Station 3, where friction also isn't a factor, the percent difference of 6.45% (which is relatively low) could have been due to a slight slant in the pendulum as it passed through the photogate or if the height that we dropped the pendulum was slightly off. To decrease this error, we could have been more precise with our measurements and also used a different device to measure the velocity to remove the slanting factor.

=<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The Law of Conservation of Energy for Mass on the Spring = <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">John Chiavelli, Andrew Chung, Amanda Fava, and Nicole Kloorfain

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Objectives:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">To directly determine the spring constant k of several springs by measuring the elongation of the spring for specific applied forces. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">To measure the elastic potential energy of the spring. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">To use a graph to find the work done in stretching the spring. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">To measure the gravitational potential and kinetic energy at 3 positions during the spring oscillation.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Hypothesis:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The spring force constant will be equal to the slopes of the graphs of Force vs Displacement for each of the springs. Furthermore, each spring will have a different k-value which will be subsequently determined by examining the slops of the graphs. The softest spring will have a small k-value while the hardest spring will have the highest k-value.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Materials and Methods:**

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<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">For this lab, we set up a ring stand with additional clamps in order to hang the various colored springs. We then measured the initial length of each spring and recorded the data. Each group member applied various masses to the springs and then recorded the change in the length. Using a meter stick we were able to measure the change in length after each mass trial. We performed this procedure to eventually determine the value of k, for four differently calibrated springs. We then graphed the results of our experimentation through a F(N) v. Distance (m) graph. This enabled us to compare and contrast the meaning of k, and calculate its value by determining the slope. As per the second clause of this lab, we used a ring stand and clamp to hang the red string. We then attached a mass of approximately 0.560 kg and taped a piece of cardboard to the bottom. This was to ensure the photogate timer underneath the setup would be able to accurately capture the motion of the spring. Using DataStudio, we then determined the value of various minimums, maximums, and equilibrium's of the period. We then used the program to find the velocity at each specific point by using the position v. time graph established.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Video Demonstrations:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">media type="file" key="Movie on 2012-02-08 at 09.02.mov" width="300" height="300"

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">media type="file" key="Movie on 2012-02-08 at 09.17.mov" width="300" height="300"

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Data A:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Data B:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Using DataStudio, it was possible to calculate the values of minimum, maximum, and equilibrium values for three periods. Then using a tangential tool, the velocity at each point was determined.

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<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> MinimumMaximumEquilibrium <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Analysis A:**

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Analysis B:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Percent Difference Sample Calculation: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

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<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Discussion Questions:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">__1. Does the data for the displacement of the spring versus the applied force indicate that the data for the spring constant is indeed constant for this range of forces?__

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Yes, the data for the displacement of the spring versus the applied force indicates that the data for the spring constant is indeed constant for this range of forces. This is because when you look at the graph as you continue to add weights, it's shape is linear. The simple fact that adding weights onto the spring creates a linear shape indicates that there is some constant. In this experiment, the constant is known as the spring constant.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">__2. How can you tell which spring is softer by merely looking at the graph?__

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">It is simple to tell which spring is softer by merely looking at its graph. To do this, you must first look at the graph of all of the displacements versus the force. In this graph, the displacement will be on the x axis while the force will be plotted on the y axis. Now, it is clear that a spring that has a great displacement is relatively soft. This is because most hard springs do not move that far when you add weights onto them. With this little piece of information, it becomes easy to tell which spring is softer by looking at the graph. The softer springs will have a smaller slope than the harder ones. In other words, the graphs for the softer springs will look more horizontal. This is because they are displaced to a larger distance and thus have to stretch out more over the x axis. The opposite is true for harder springs. Since they are not displaced by a great distance their graphs are more vertical.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">__3. Describe the changes in energy of the hanging mass, beginning with it starting at rest, you pulling the mass down and then releasing it, and then the mass cycling through one complete period.__

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The changes in energy for the hanging mass at various times is dictated by it's location. For instance, at the lowest point the hanging mas has only gravitational potential energy and elastic potential energy. This is, of course, dependent on the fact that we set the zero for GPE at a point lower than the distance that the spring travels at it's lowest height. While it is at equilibrium, it's energy is made up of kinetic energy, gravitational potential energy, and elastic potential energy. Finally, at it's highest point, also known as it's maximum displacement, the energy components include only gravitational potential energy. It is important to note that the total energy at the minimum and maximum displacement as well as at equilibrium have to be equal. This is due to the Law of Conservation of Energy. Thus, at each of these points the values for gravitational potential energy, kinetic potential energy, and elastic potential energy change depending on where exactly the weight is at that time.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Conclusion:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">For part one of the lab, we were very close to accuracy with our results. This lab had very little room for error as long as you accurate found the length of the springs compressed and uncompressed with a certain weight on it. For each spring we were within the 10% error range it provided on the box. Our only source of error could have been recording the lengths of the springs because it was difficult to balance the meter stick in the air next to the spring. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The results for the second clause of the lab were very good. After performing three trials and averaging the results, followed by calculation, the percent difference between the three stages of energy motion was 9.45%. This is reflective of sound results and laboratory experimentation. A possible source of error could have been the result of the photogate timer misrepresenting the actual motion of the mass spring. This would cause discrepancies on the position v. time graph, in turn affecting the values of height, x, and velocity.

=<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">__Roller Coaster Project__ =

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**//Video and Views/Pictures/Diagrams://** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

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<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**//Data//**: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">//Theoretical speed was determined by plugging in the theoretical KE values (and .016 as the mass) into the equation: //

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">//Sample for Percent Error Calculation:// <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**//Video//**: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">media type="file" key="IMG_0364.mov" width="300" height="300" <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**//Sample Calculations://** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**//Analysis://**
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">theoretical amount of energy and power to get roller coaster rolling
 * [[image:Screen_Shot_2012-03-09_at_4.43.51_PM.png]]
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">minimum speed requirement at top of vertical loop
 * [[image:Screen_Shot_2012-03-09_at_4.44.07_PM.png]][[image:_.png]]
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">minimum height requirement of first hill based on (2)
 * [[image:Screen_Shot_2012-03-09_at_5.30.13_PM.png]]
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">energy dissipated at end of ride to stop roller coaster
 * [[image:Screen_Shot_2012-03-09_at_4.44.24_PM.png]]
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">theoretical spring system to stop roller coaster car if brakes fail
 * [[image:Screen_Shot_2012-03-09_at_4.44.32_PM.png]]

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**//Discussion://**

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Because of the Energy Conservation Law, we know that energy cannot be created or destroyed, and therefore must remain the same throughout our entire coaster. Though, because of sources of error and because we ignored friction and air resistance, it is understandable that our total energy did not remain constant from start to finish. The friction between the marble and the track serves as the work, as it is a force that causes displacement; in our coaster, it is opposing the direction of motion and is therefore negative work. Since we did not take into account the effect of work when finding our theoretical values, the total energy at any given place would not equal the total energy at the beginning of the coaster. If we had taken work into account, the total energy would always be the same, as energy would just be transferred from GPE to KE and also work. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Newton’s three laws of motion are also apparent in our roller coaster. According to Newton’s first law, an object will stay in motion unless acted upon by an unbalanced force. This is shown in our roller coaster, since the marble continues to move down the roller coaster from the initial drop, until it reaches the end where we have a stop barricade (but would actually be a spring stopping system). Though, there were other forces acting on the marble, such as gravity, friction, and even normal, which is why the marble did not move at a constant velocity. As described by Newton’s second law, these forces, combined as a total net force, acting on the marble cause acceleration. Newton’s third law can also be applied to the roller coaster, since it states that every action has an equal and opposite reaction. The marble represents a cart holding people; these people have a normal force exerted on them from the seat, and equally, they exert a normal force on the seat. Because of the normal force they exert on the seat, they are able to stay within the seat during vertical loops. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Acceleration was a key part of this roller coaster. Because the slopes were constantly changing for each drop and through loops, the acceleration was also constantly changing. Therefore, to find acceleration at each point, we had to use our knowledge of kinematics. We had to measure the distance between intervals, then we also used the initial and final velocities at those intervals (which we had found using photogate timers and the mass of the marble) in the equation Vf=Vi+2ad to find the acceleration. We also used our knowledge of centripetal forces to calculate acceleration. For example, at the vertical loop, we found the minimum velocity necessary to allow the marble to pass through the loop. Normal force and weight were the two forces acting on the marble, so we used the equation F=m(v^2/r). The concept of apparent weight can be applied here. Apparent weight is what the weight apparently feels like at any given position using normal and gravity forces. Therefore, an object may feel like it has no weight at certain points, such as the top of a hill, when gravity and normal are in different directions. Conversely, at the bottom of a hill, the apparent weight would be greatest. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Although our prototype starts with the initial incline, we found the theoretical power necessary for the marble to climb from the bottom to the top of the initial incline. Power is the rate at which work is performed, so therefore it can be found by dividing the amount of work and the total time. Since this is a theoretical incline, we decided that the journey to the top of the incline would take 30 seconds. Since gravitational potential energy is the only energy at the top of the incline, we divided that by 30 seconds to get a power ration of 0.00459 Watts. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">In addition, our prototype is also lacking a stopping system. The actual roller coaster would have a spring system at the end of the coaster to stop the cart in the event that the breaks fail. We used Hooke’s Law to find the elastic potential energy for the spring system. We decided that we wanted the compression distance of the spring (x) to be 0.15 meters, and in combination with the total energy dissipated at the end of the roller coaster (0.128 J), we used Hooke’s Law (EPE=k*x) to find the theoretical spring constant (k) of 11.38. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Our roller coaster can be determined as pretty safe. The accelerations at each point were pretty low, with most being between 0-2 g's. The only safety issue would be in the horizontal loop and the back curve. The marble gains a lot of speed from the first loop and the drop that the acceleration at the back curve is 4.92 and is 4.01 at the horizontal loop. Since the maximum g's of acceleration that a person can endure is 4 g's, these spots would probably cause light-headedness in the passenger, and may even be high enough at the back curve to cause blacking out. Though, overall, the roller coaster would be classified as pretty safe though thrilling. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">There were multiple sources of error throughout this roller coaster, which explain a lot of the uncertainties in our data (such as the total energy not being the same throughout the entire coaster). The prototype was made out of paper tracks, and although we used supports, the stability was still not guaranteed. Since the paper only got more flimsy as we kept making new additions to the prototype, it is highly probable that the tracks moved slightly between trials which would throw off our data. We could fix this problem in the future by using a different material, something that provides more support than paper, such as wood or metal, as would be used for the real roller coaster. In addition, the estimation of the last significant figure when measuring height could have also thrown off our data and could be corrected by using a measuring tool more exact than a meter stick. In addition, the photogate timers used to find the time between intervals and at each point could have provided some of the error. Since we had to place these timers on stands and then on the prototype, it is possible that they caused the tracks to move or that the time could have been the track moving in the photogate instead of the ball. To correct this source of error, we would use a tool other than photogate timers that can more precisely determine the time and ultimately the velocity at each point. Another obvious error was the lack of taking air resistance and friction into consideration, as mentioned previously. This could be corrected by finding/determining the coefficient of friction between the track and the marble and the air resistance at each point, then using them with the Work-Energy equations to find the total energy throughout the coaster. All these mistakes explain our high percent errors, which go as high as 44% at the horizontal loop, where these factors would seem most prominent. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Despite our high percent errors, our coaster should still be chosen for the amusement parK. We have created a safe but thrilling roller coaster, and we know the public with love to ride the Crash N' Burns!

=<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Elastic and Inelastic Collisions Lab =

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Objective: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">What is the relationship between the initial momentum and final momentum of a system? Which collisions are elastic collisions and which ones are inelastic collisions?

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Hypothesis:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The initial momentum should be the same as the final momentum for elastic collisions. This is not true for inelastic collisions.

Methods and Materials: We placed two carts on a dynamic track, with motion sensors at both ends. Data Studio, where a USB connected the sensors to our computers, was used to collect the velocities. Five different collision scenarios were used, and each was conducted with various masses and multiple trials.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">media type="file" key="Movie on 2012-03-14 at 08.42.mov" width="300" height="300"

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Data:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Sample Calculations/Graphs: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Analysis:
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Is momentum conserved in this experiment? Explain, using actual data from the lab.
 * 2) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Momentum is not conserved in all of the experiments except the one where the carts stick together as well as the explosion. The initial kinetic energy and the final kinetic are not equal because it was not an overall constant amount of energy. In an explosion, the carts start at rest and then has a final velocity. There will always be a 200% difference because it is the absolute value of final of momentum minus intial over the average times 100. These are the inelastic collisions which would have the higher percent differences. The elastic collisions have very similar initial and final kinetic energy and are conserving kinetic energy therefore have a low percent difference which was common in the rest of the experiments.
 * 3) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">When carts of unequal masses push away from each other, which cart has a higher velocity? Explain why this is.
 * 4) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The lighter massed cart has a higher velocity. When the cart with the greater mass pushes against the cart with the smaller mass in the collision, the force exerted upon the smaller one is greater then the force exerted on the larger one.
 * 5) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">When carts of unequal masses push away from each other, which cart has more momentum?
 * 6) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The cart with the larger mass because momentum equals mass times velocity therefore the greater the mass, the greater the momentum.
 * 7) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Is the momentum dependent on which cart has its plunger cocked? Explain why or why.
 * 8) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Momentum is not dependent on which cart has the cocked plunger. The two factors are mass and velocity. The plunger should have an equal affect on the two cars because it has the same force acting upon it, which would make the plunger independent of which cart it is attached to.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Conclusion:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">This lab was unique in that various tasks were split up amongst the groups in the class. Our group completed an experiment in which the two carts ran into each other and stuck, by way of Velcro. Our hypothesis thus stated that this collision would be an elastic one. From what we have learned in class, we have come to know that in scenarios such as this one, energy is conserved rather than lost. There are, essentially, two types of collisions. Elastic ones and inelastic ones. In elastic collisions, energy is conserved while in inelastic ones, energy is lost. We hypothesized that this would be an elastic collision and energy would subsequently be conserved. Further analysis and experimentation proved us to be right. When comparing the initial momentum and final momentum of the various trials that we ran, it became clear that the values were very similar. We ran several different trials, continually varying the amount of weights that we used. For instance, on some trials we placed two weights on one cart and one on the other, and vice versa. As one can see if one looks at the data above, each cart weighted something different on each trial. We did this to cover as many bases as possible. In this lab, we also selected a specific side as being a negative direction and the opposite side being positive. Since negative/positive values in physics are relative, we thought that it would be prudent to stick to a side.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">We discovered that the initial momentum and the final momentum of our system were very similar. This indicates that the collision is an elastic one, as momentum should theoretically not change in elastic collisions. The reason for the difference in our values is simple: sources of error. As with any experiment conducted by humans, there are bound to be some errors. This was the case in our lab. For instance, the carts could have stuck to the ramp, changing the velocity. The masses of the carts could have also been incorrect or not exact enough, leading to an incorrect momentum calculation. It is also possible that we could have selected the wrong point on DataStudio or accidentally triggered the motion detectors. If we had to do this experiment again, we would measure all of the items used very carefully. We would also see if we could find a smoother ramp.

=<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Ballistic Pendulum Lab =


 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Objective: **
 * What is the initial speed of a ball fired into a ballistic pendulum?


 * Hypothesis:**
 * The initial speed of the ball should be about the same for each method. The photo gate method should be the most accurate and the Law of Conservation of Momentum should be the least.

In order to find the initial velocity of the launcher, there were several ways to test the result. First, push the ball into the launcher as far back as it can go therefore it is in the highest range. For one experiment, place a photogate timer right under the launcher exit which will generate the time. Next experiment is using a kinematics problem through projectiles. Launch the ball to trial where it will land. Place carbon paper where the ball will land. Find the height and distance of where the ball started (in the launcher) to where it landed (the floor). Last trial is using the Law of Conservation of Momentum is launching the ball into a ballistic pendulum that is in rest. See the change in the angle to help find the height of how hight the pendulum moved from the velocity of the ball.
 * Methods and Materials:**


 * Videos and Photos:**

Law of Conservation of Momentum media type="file" key="Movie on 2012-03-21 at 08.24.mov" width="300" height="300"

Photogate Timer media type="file" key="Movie on 2012-03-21 at 08.47.mov" width="300" height="300"

Kinetmatics media type="file" key="Movie on 2012-03-21 at 08.51


 * Data:**




 * Sample Calculations:**


 * Analysis:**

1. In general, what kind of collision conserves kinetic energy? What kind doesn't? What kind results in maximum loss of kinetic energy?
 * In general, elastic collisions conserve kinetic energy. This differs from inelastic collisions, which do not. In order for there to be a maximum loss of kinetic energy, all of the kinetic energy must be lost. Such an event occurs when a car crashes into a wall.

2. Consider the collision between the ball and pendulum.
 * 1) Is it elastic or inelastic?
 * It is inelastic.
 * 1) Is energy conserved?
 * No, energy is not conserved because the collision is inelastic.
 * 1) Is momentum conserved?
 * Momentum is conserved due to the Law of Conservation of Momentum.

3. Consider the swing and rise of the pendulum and embedded ball.
 * 1) Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height?
 * Energy is not conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height.
 * 1) How about momentum?
 * Momentum is conserved.

4. It would greatly simplify the calculations if kinetic energy were conserved in the collision between ball and pendulum.
 * 1) Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum.
 * 2) [[image:Screen_shot_2012-03-22_at_12.32.08_AM.png width="163" height="119"]]
 * 3) [[image:Screen_shot_2012-03-22_at_12.30.44_AM.png width="179" height="120"]]
 * 4) [[image:Screen_shot_2012-03-22_at_12.33.27_AM.png width="212" height="84"]]
 * 5) What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy.
 * 6) [[image:Screen_shot_2012-03-22_at_12.35.04_AM.png width="198" height="131"]]
 * 7) According to your calculations, would it be valid to assume that energy was conserved in that collision?
 * 8) No. There was an 85.25% loss as the analysis above shows. This means that energy was //not// conserved during the collision, indicating an inelastic collision.
 * 9) Calculate the ratio M/(m+M). Compare this ratio calculated in part (b). Theoretically, these two ratios should be the same. State the level of agreement for these two quantities for your data.
 * 10) [[image:Screen_shot_2012-03-22_at_12.37.09_AM.png width="394" height="72"]]
 * 11) We received a value of 78.9% for this calculation. This differs from our percent loss of 84.25%. This is most likely due to the fact that there were several issues with our data collection of the pendulum. Specifically, the initial velocity seems to be a bit off from what other people in our class got.

5. Go to http://higheredbcs.wiley.com/legacy/college/halliday/0471758019/simulations/sim13/sim13.html and select "Ballistic Pendulum" from the column on the left. What is the effect of increasing the mass of the ball? What is the effect of increasing the pendulum mass? Try it. (NOTE: You have to read "Student Notes" first before you can run the simulation.)
 * When you mass of the ball increases, the initial velocity increases as a result. This results in an increase in the height of the pendulum and of theta, the angle. This differs from the result that occurs when you increase the mass of the pendulum. Increasing the mass of the pendulum decreases the height and also makes theta, the angle, smaller.

6. Is there a significant difference between the three calculated values of velocity? What factors would increase the difference between these results? How would you build a ballistic pendulum so that the momentum method gave better results?
 * There is not a significant difference between the three calculated values of velocity. As seen above, the values are 5.190, 4.894, and 4.927. The percent difference value for all of these values is within 5%, which is reasonably consistent. There are other factors that would increase the difference between these results. For instance, it is necessary to account for air resistance. The air could have had an impact on the values that we calculated, creating different results. The ballistic pendulum could have also been at fault. For instance, the ball could have hit something on its way into the pendulum that slowed it down. There are simply too many extraneous variables to take into account when doing a lab like this to expect a 0% difference and perfect results. If I had to build a ballistic pendulum so that the momentum method gave better results, I would design it so that the transfer from the ball to the pendulum was extremely efficient.

The average velocities for the different components of the lab. For the pendulum, the average velocity was 5.19 m/s. For the kinematics, the velocity of the ball was 4.829 m/s. For the photo gate, the velocity of the ball was 4.927 m/sOur hypothesis stated that the initial speed of the ball should be about the same for each method. We believed the photo gate method should be the most accurate and the Law of Conservation of Momentum should be the least. We were correct our percent difference was smallest or most accurate for the photogate and the largest was when using the Law of Conservation of Momentum. Therefore our hypothesis was correct; the photo gate strictly provides time. It has the smallest amount of factors in the equation to create poor results. Our velocity for that was 4.88 which was closest to the the results we achieved from the experiment 4.927 m/s. Our highest percent difference was 4.04% which is very successful. The part of lab where error could have occurred was the launcher shooting the ball at different ranges. Also, when we use the launcher for too many trials in a short amount of time the launcher will change its speed so by the end that could have been a source of error. Regardless the launcher may not have been shooting the ball at a consistent speed. The reason that our pendulum may have been flawed is because the ball have been launched from different ranges. For kinematics, the launcher was not clamped to the table so it may have moved a little bit changing the dimensions of distance and height of where the ball landed. Our trials for each part of the lab were very consistent. We seemed to be getting very good results. These concepts could be seen in car crashes, explosions and even in a game of billiards. All include the element of momentum. They all involved mass, velocity, and direction. They also display how kinetic energy and momentum are either conserved or lost in circumstances. If a car hits a stationary car, the momentum transfers from the car in motion to the stationary one which is like one billiard ball hits another as well as the ballistic pendulum.
 * Conclusion:**