Group5_2_ch6

Jake Aronson Dani Rubenstein Jessica Smith Michael Solimano
 * Members:toc**

=**Law of Conservation of Energy Lab**= Jake--Wiki set-up (Objectives; Materials and Methods; Procedure); Calculations and Analysis Dani--[absent during lab] Jessica--Calculations and Analysis; Conclusion Michael--Data; Conclusion
 * Tasks:**


 * Objectives**
 * 1) If the cart starts at the top of the ramp, what is its speed at the position of the photogate?
 * 2) What is the speed of the ball when it leaves the launcher at short range?
 * 3) What is the speed of the pendulum at the bottom of its path, if released from h=20 cm?
 * 4) What is the highest point the ball will reach when released from the top of the shorter incline?
 * 5) What is the speed of the ball when vertically launched at short range?
 * 6) What is the speed of the ball at the top of the loop?

Our goal is to achieve data at each station that prove that the total initial energy is equal to (or, at least, explainably close to) the total final energy. The Law of Conservation of Energy states that energy cannot be created or destroyed, but it can change forms. If our data yield calculations for initial energy and final energy that are close, then our lab will have been successful in quantifying the Law of Conservation of Energy.


 * Materials and Methods**


 * 1) At Station 1, we attached a block to a cart at the top of an incline, and plugged the DataStudio USB cable into the computer (click "Recordable Timer"). We released the cart and let it roll down the incline and through the photogate, which recorded a "Time in Photogate" measurement on DataStudio.
 * 2) At Station 2, we loaded a ball into the launcher (aimed at 0 degrees) at short range, and plugged the DataStudio USB cable into the computer (click "Recordable Timer"). We pulled the trigger to the launcher and let the ball travel as a projectile through an initial photogate at the mouth of the launcher, and a final photogate just above the ground, the latter of which recorded a "Time in Photogate" measurement on DataStudio.
 * 3) At Station 3, we held a pendulum at 20 centimeters above its vertically circular path, and plugged the DataStudio USB cable into the computer (click "Recordable Timer"). We released the pendulum and let it swing down its path through the photogate at the bottom of its path, which recorded a "Time in Photogate" measurement on DataStudio.
 * 4) At Station 4, we held a ball at the top of the shorter incline on a two-incline ramp. We released the ball and let it roll down the shorter incline and up the longer one, until it reached its maximum height on the taller incline, where we marked and measured the height.
 * 5) At Station 5, we loaded a ball into the launcher (aimed at 90 degrees) at short range, and plugged the DataStudio USB cable into the computer (click "Recordable Timer"). We pulled the trigger to the launcher and let the ball travel upwards through the photogate, which recorded a "Time in Photogate" measurement on DataStudio, and measured the maximum height of the ball's path.
 * 6) At Station 6, we held a ball at the top of an incline that connected to a loop, and plugged the DataStudio USB cable into the computer (click "Recordable Timer"). We released the ball and let it travel down the incline and through the loop and the photogate at the top of the loop, which recorded a "Time in Photogate" measurement on DataStudio.

Also, at each station, we measured the height and mass of the system.

media type="file" key="AronsonSmithSolimanoGroup10.m4v" width="300" height="300"
 * Procedure (video)**
 * Data**

Class Data Station 1


 * Calculations and Analysis**



The purpose of these stations was to analyze the initial total energy and the final total energy of different experiments. In theory, these should have been equal to each other, which we hypothesized, as according to the law of conservation of energy, energy cannot be created or destroyed, only transformed. We used this principle to analyze the initial energy and final energy in different settings, which allowed us to generate a percent difference for each station. The initial and final energies were generated by first establishing equations, to which numbers could be plugged in to analyze the closeness of the initial and final energies. It was found that our hypothesis was generally wrong, as the final total energy tended to be less than the initial total energy. This is probably because of the set up of our labs. Many of the stations could have introduced friction onto the mass being analyzed, which would have reduced the final total energy. Had we accounted for all of the forces we could have generated more accurate results. An example is station 3, where our percent difference was 36.276%. This appears to be a high percent difference, but was not the highest, as many of the stations had even higher percent differences. As stated previously, this high percent difference was probably because of unnaccounted forces. In this station, friction could have slowed our ball, which would have ultimately reduced our final energy and accounted for the difference.
 * Conclusion**

=Lab Spring Force Constant= Members: Michael Solimano Jessica Smith Dani Rubenstein Jake Aronson (absent)

1. To directly determine the spring constant k of several springs by measuring the elongation of the spring for specific applied forces. 2. To measure the elastic potential energy of the spring. 3. To use a graph to find the work done in stretching the spring. 4. To measure the gravitational potential and kinetic energy at 3 positions during the spring oscillation.
 * Objective:**

Part A: The spring force constant will be equivalent to the slope of the graph of applied force vs displacement. Part B: The point of equilibrium will be the point when the spring is at rest. The minimum displacement is the point where the spring is pulled to. The maximum displacement is the point where the spring is at a maximum after being released. These points and there accompanying velocities will allow us to analyze the changing GPE and KE at different points. However, the total amount of energy at each of these points will be equal.
 * Hypothesis:**

For Part A, we first set up a set of red, blue, and white, springs on a rod and used clamps to attach the springs to it. Then, we made an Excel spreadsheet in order to keep track of the spring we were using, the mass that was attached to the spring, and the displacement that the masses caused the spring. Next, we used a mass set to attach different masses to each spring and used a meter stick to measure the displacement that each individual mass caused. We recorded this information in our spreadsheet and we were able to calculate our experimental spring constants by creating a graph of Force vs. Displacement. For Part B of the lab, we also attached a spring to a rod. However, at the end of this spring we only used one mass that was taped to a piece of cardboard in order for it to be detected by the motion sensor. The motion sensor, which we used through Data Studio, was able to give us information such as the velocity at the object's maximum displacement, minimum displacement, and equilibrium. Then, we recorded this data in an Excel spreadsheet and examined our results.
 * Methods and Materials:**

__**Part A**__


 * Procedure:**


 * Data:**

Our initial mass was .250 kg, but we zeroed it and changed all of the other values accordingly Data Table Part A:



Sample Calculation for Force: (Red Spring Mass of .1 kg)
 * Sample Calculations:**
 * Percent Error for Spring Force Constants:**

We chose to use percent error for this part of the lab because we were finding the error between our experimental results and the theoretical results that were found on the box of the spring set. If we were comparing our results to a class average, for example, we would have used percent difference, however, we were not doing that.


 * Percent Difference for Spring Force Constants:**

__Red Spring__: Percent Difference for Red Spring = 3.44% (Sample Calculation Below) __Blue Spring__: Percent Difference for Blue Spring = 1.32%

__White Spring:__ Percent Difference for White Spring = 2.18%

For these comparisons we chose to use percent difference because we were comparing our individual results to the average results of the entire class.

__**Part B**__

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 * Procedure:**


 * Diagram of Part B:**


 * Graph of Part B:**


 * Data for Part B:**


 * Sample Energy Calculations:**

Order of Percent Differences Below: Equilibrium, Minimum Displacement, Maximum Displacement

For this part we chose to use percent difference to evaluate our results because we were comparing the individual value at each position to the average values of all three positions. We did this because all three positions were supposed to have equal amounts of energy so we were finding the difference of each one compared to the average.

1. Do 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? Yes, the data we calculated does imply a constant spring force. We knew this because the graph of our data was linear. As we added weight, or applied force to our spring, its measured displacement formed a linear graph. Thus, the spring force constant must have been a constant in order for a linear graph to occur.
 * Discussion Questions:**

2. How can you tell which spring is softer by merely looking at the graph? The less steep the slope, the softer the spring would have been. This is because our graph measured the applied force, which was in uniform intervals for our three springs, on the y axis, and the displacement of the spring on the x axis. Thus, the more the spring was displaced, or the softer it was, the more it would have displaced along the x axis as applied force increased. Thus, the spring with the least steep curve would have had the most displacement by adding weight, and would be the softest.

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. The total energy of part 2 should ideally be identical throughout the period of the mass bouncing. As it passes through equilibrium, the mass has Kinetic Energy, GPE relative to its minimum, and EPE because the spring is still coiled.. At its minimum displacement, the mass has EPE because the spring is stretched, as well as GPE. At its maximum displacement, it will have GPE. It will not have KE as it is not in motion, and no EPE because the spring is not coiled.

In Part A of this lab, we were mainly aiming to determine the spring force constant in each individual spring. We hypothesized that the spring force constant that was found on the box was going to be equal to the slope of the trend line found by creating a Force vs. Displacement graph of each spring. By using varying masses and measuring the displacement that each of these masses made, we were able to use our data to create a graph showing the Force vs. Displacement of each spring. From this graph, we looked at the equation of each trend line and found our results. For the red spring, our graph showed the spring force constant to be 25.068. For the white spring, our graph showed the spring force constant to be 41.087. Finally, for the blue spring, our graph showed the spring force constant to be 30.446. We immediately were able to tell that our results were accurate, since the theoretical spring force constants on the box were 25, 40, and 30, respectively. However, we used both percent error and percent difference to prove that these results were in fact as accurate as we thought. First, we did percent error to determine the error between our experimental results and the theoretical results on the box. For the red, white, and blue springs our percent errors were as follows: .272%, 2.65%, and 1.48%. All of these percent errors fall within a 5% range, showing that our results were extremely accurate. Also, we calculated the percent difference of our results compared to the average results of the class. For the red, white, and blue springs our percent differences were as follows: 3.44%, 1.32%, and 2.18%. Clearly, both our percent errors and percent differences show that we had very successful results in this part of the lab. This was so due the fact that this lab left little room for error. It simply consisted of adding up different masses and measuring displacements, and by carefully using the meter stick we were able to obtain very good results. Our minor errors, however, could have been due to the fact that the spring could have still been slightly moving when we measured it, evidently altering the measured displacement. For Part B, we were mainly aiming to determine that the total amount of energy at the maximum, minimum, and equilibrium points were the same. We hypothesized that these results were going to be close to exactly the same. After calculating the velocities and heights at each of these points, we were able to use the Law of Conservation of Energy equation to prove that the total energies were equal at all of these points. After completing this calculation, we found that we had results that were very close, but they were not exactly equal to each other. To evaluate these results, we averaged the three numbers together and found the percent difference from equal individual point (maximum, minimum, and equilibrium) compared to this average. Our percent differences for equilibrium, minimum, and maximum were as follows: 4.28%, 3.35%, and .739%. Once again, our results for this part of the lab were extremely accurate. They all fell within a 5% range, evidently proving that we were successful in obtaining accurate results. However, one source of error for this part of the lab could have been the fact that the sensor has the best results when the object is a certain distance away. If we moved the object slightly closer to the senor when releasing it, we would most likely have had even more accurate results. To fix this source of error, we would have made sure the object was closer to the sensor when we released it. This lab pertains to many real life situations that engineers and physicists have to face every day. For example, when creating a roller coaster ride, the spring force constant must be determined in order to ensure that the ride will be safe for the people on it. If the spring force constant was too high, the cart might reach an acceleration that is too high for a person to withstand. If it is too low, the acceleration might not be high enough in order for the card to make it through a loop.
 * Conclusion:**

=Roller Coaster Project=

//By Michael Solimano, Dani Rubenstein, Jessica Smith, Jake Aronson//


 * Pictures**


 * Video**

media type="file" key="AronsonRubensteinSmithSolimano (Group 10) Video.mov" width="420" height="420"


 * Experimental Data**




 * Theoretical Data**




 * Percent Error**

Minimum Speed Around Vertical Loop Minimum Height Requirement Based on Minimum Speed Around Vertical Loop Horsepower of Motor
 * Sample Calculations**
 * Discussion**

Our roller coaster demonstrates energy conservation, power, Hooke's Law, acceleration, Newton's Laws, and gravitation and apparent weight.

As we have learned in class, the Law of Conservation of Energy states that energy (measured in Joules) cannot be created or destroyed, but it can change forms (for example, from gravitational potential energy to kinetic energy). This concept was demonstrated throughout our project. Based on the gravitational potential energy at the top of our initial drop, we calculated that the theoretical total mechanical energy at every point on our roller coaster should have been 0.293 Joules. However, based on our photogate measurements, we calculated that the total mechanical energy at every point was different. Realistically, sources of error (which are discussed further down) affected the total mechanical energy, and some energy was dissipated as work. A sample calculation for energy dissipated as work is shown above (see "Energy dissipated..."). The total mechanical energy that we calculated to be closest to 0.293 Joules was at the bottom of hill 3, where the total mechanical energy was 0.248 Joules.

Next, we have learned in class that power (measured in Joules/second, or horsepower) is the rate at which energy is used or dissipated. Our roller coaster did not employ a motor to provide power to bring the marble to the top of the initial drop, so we calculated the power that would have been required if we had used a motor. We thought that 30 seconds would be a suspenseful ride to the top of a tall roller coaster, so we calculated the amount of work done divided by 30 seconds. Because our roller coaster was so small (relative to a real coaster, like El Toro), the power for our hypothetical motor would have to be 1.31e-5 horsepower. The calculation for power is shown above (see "Horsepower of Motor").

In addition to calculating the power of a hypothetical motor, we used Hooke's Law to calculate the spring force constant for a hypothetical spring force system (which we would use to stop our roller coaster if the brakes failed). Hooke's Law states that the amount of force on a spring is proportional to the amount of stretch or compression (elastic potential energy is equal to one-half times the spring force constant times the distance of stretch). First we used kinematics to solve for the distance needed to stop safely, then we set the elastic potential energy equal to the amount of work that we had calculated for that point on our roller coaster. Using the plug-and-chug method, we solved for a theoretical spring force constant that would serve as that of the theoretical spring force system. The calculation for spring force constant is shown above (see "Theoretical Spring System...").

The marble also experienced acceleration at every point on our roller coaster, whether positive or negative. To calculate these accelerations, we used our measurements for velocity and change in distance from the top and bottom of every point on our roller coaster. We set up kinematics equations and solved for the accelerations. A sample calculation for acceleration is shown above [see "Acceleration (at bottom...)"]. Based on our calculations of acceleration, we can conclude that our roller coaster is very safe for a human to ride at every point. The maximum acceleration that the human body can tolerate is 4g, or 39.2 meters per second-squared. The maximum number of "g"s that we calculated on our roller coaster is 2.514, so the human body could withstand our roller coaster. Though our roller coaster would not be as much fun as El Toro, it would be certainly be safer for the human body.

Also, we observed Newton's first and second laws of motion in action at many points on our roller coaster. We have learned that Newton's first law of motion states that a body in motion will stay in motion until acted upon by an opposing force. Wherever we observed negative acceleration, we observed this law of motion. The marble would have continued to gain velocity, but it was slowed by the hills, turns, loops and end stopper. We have also learned that Newton's second law of motion states that the net force of an object is equal to the object's mass times its acceleration. More specific to this project, we have learned about centripetal motion: the net force of an object is equal to the object's mass times the quantity of its velocity-squared divided by the radius. We used Newton's second law, and its application in centripetal motion, to calculate the minimum velocity (which is equal to the square root of the radius times "g") for the marble to travel around each loop, and the acceleration (which is equal to the velocity-squared divided by the radius) of the marble around each loop. Sample calculations for minimum velocity and acceleration are shown above (see "Acceleration About a Loop" and "Minimum Velocity Around Vertical Loop").

Finally, the marble was affected by gravitation throughout its roller coaster ride, and it experienced apparent weight at a few points on the roller coaster. We accounted for the force of gravity in most of our calculations, whether in gravitational potential energy (which is an object's mass times its height above 0 times "g") or weight force (which is equal to an object's mass times "g"). Gravity was always acting on the marble's vertical axis (that is what kept it from flying off of the tracks). Specific to this project, an object's apparent weight is the force that the object exerts on the surface beneath it. The marble felt an apparent weight, different from its weight, whenever there was a normal force acting vertically upwards against the downwards force of gravity. For example, the marble felt no apparent weight at the top of the hills, because there, the normal force was equal to 0. This lack of apparent weight is known as "weightlessness," which occurs when gravity is the only force acting on an object. The marble also would have felt weightless at the top of the vertical loop, if it was going through the loop at the minimum speed required to complete the loop. When the marble is weightless, and its apparent weight is equal to 0, its true weight is equal to the force of gravity: mass times "g."

A few sources of error affected our calculations. First and foremost, our roller coaster could have been more stable. If we had used many more supports for the tracks and columns, the marble would have had a smoother ride on the roller coaster, and the total mechanical energy at every point on our roller coaster would have been much closer to the initial gravitational potential energy. Also, fewer bumps along the ride would have made our photogate timings more precise. Second, our photogate timings were subject to human error. Because we had to hold the photogate in our hands while measuring the velocity of the marble at each point on our roller coaster, the measurements could have been affected by our hands shaking, and the photogate might not have measured the marble at the right time. If we had used stands and poked holes so the marble had a clear path through the photogate laser, our measurements would have been much more precise. Third and lastly, the construction of our roller coaster changed after every trial. The roller coaster was made out of paper, so it was malleable and prone to bends and tears. These bends and tears could have affected our photogate measurements, so we could not get accurate results for our calculations. These sources of error caused our percents error to be very high. The average percent error that we calculated was 252.262 percent error, indicating that our data were very different from the theoretical values for each position on the roller coaster. However, our project overall was a success, because the marble traveled through the entire roller coaster successfully many times. In fact, if you watch our video, you can see that it did so three times in a row (see "**Video**").

Thank you for reading about our roller coaster project!

=Momentum Lab=

Task A- Jess Smith Task B-Michael Solimano Task C- Jake Aronson Task D- Dani Rubenstein

1. What is the relationship between the initial momentum and final momentum of a system? 2. Which collisions are elastic and which ones are inelastic collisions? The purpose of this lab is to use our knowledge of the law of conservation of momentum to analyze our system, which was an example of an explosion, for its initial and final momentums, which should be equal. We further will test its initial and final kinetic energies to determine whether or not the collision was elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved).
 * Purpose/Objective**

1. The initial and final momentums of our system will be equal, because of the Law of Conservation of Momentum, which states that momentum is conserved. 2. The collisions that will be elastic are the ones that conserve kinetic energy, and those that are inelastic will fail to conserve in their reactions the kinetic energy.
 * Hypotheses**

The materials in this lab include a level track, two carts, weights, distance sensors, and laptops.
 * Materials and Methods**

In this lab, two carts were placed next to each other on a level track, with two sensors that measured their distances on each end of the track. A spring in one cart was activated and the carts "exploded" apart. The sensors picked up their changing distances, which were used to analyze their momentum. Different weights were added to the carts to test varying conditions in our experiment.
 * Procedure**

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 * Video**


 * Group Data**




 * Class Data**


 * Analysis Questions**


 * 1) Is momentum conserved in this experiment? Explain, using actual data from the lab.
 * 2) Momentum was not conserved in our experiment. According to the Law of Conservation of Momentum, a system's initial momentum is equal to its final momentum, as defined by the equation m1v1i + m2v2i = m1v1f + m2v2f . Our group's experiment was an explosion of two push carts, which were initially connected to each other while at rest. Because the initial velocity for both carts was equal to 0, our system's initial momentum, or m1v1i + m2v2i, was equal to 0 as well. However, our final velocities ranged from 0.09 m/s to 0.45 m/s for one cart, and from -0.19 m/s to -0.44 m/s for the other cart, so our system's final momentum ranged from -0.05478 kgm/s to 0.03939 kgm/s. Because our system's initial momentum was not equal to its final momentum, our collision was inelastic, meaning that momentum was not conserved.
 * 3) When carts of unequal masses push away from each other, which cart has a higher velocity? Explain why this is.
 * 4) When carts of unequal masses push away from each other, the cart with the smaller mass has a higher velocity. The equation for the Law of Conservation of Momentum is m1v1i + m2v2i = m1v1f + m2v2f, meaning that initial momentum equals final momentum, but the initial velocities for both carts in our explosion was 0, so our equation changes to m1v1f = -m2v2f. The Law of Conservation of Momentum tells us that the cart with the smaller mass would have to yield the same momentum as the cart with the larger mass, so its final velocity (v1f) would have to be larger to balance its smaller mass, and the cart with the larger mass would have to yield a final velocity (v2f) smaller than the other cart's final velocity.
 * 5) When carts of unequal masses push away from each other, which cart has more momentum?
 * 6) When carts of unequal masses push away from each other, both carts have equal momentum because momentum is equal to mass times velocity. The cart with the smaller mass would have a greater velocity, and the cart with the larger mass would have a smaller velocity, so the equation would be balanced. For example, if two carts of masses 0.5 kg and 1 kg pushed away from each other, then the ratio of the final velocities would remain 2 to 1, respectively. This means that the final velocities could be 1 m/s and 0.5 m/s, respectively, or even 2 m/s and 1 m/s, respectively. The equation would look like this: (m1)(2v2f) = m2v2f.
 * 7) Is the momentum dependent on which cart has its plunger cocked? Explain why or why not.
 * 8) The momentum is not dependent on which cart has its plunger cocked; the momentum is dependent upon the masses of the carts. Going back to an earlier lesson, we know that Newton's Third Law of Motion states that the force of one object contacting another is always equal to the force of the other object on it, so the cart pushed off of by the released plunger would experience just as much force as the cart pushing off with the plunger (the one with the cocked plunger). In fact, if the plunger were switched to the other cart, both carts would still experience the same force from the released plunger. Because the force acting on both objects would be the same, we know that the resultant velocities, and therefore the resultant momentums, would depend on the carts' masses: the same force would have a greater impact on the cart with the smaller mass, making its velocity greater than that of the cart with the larger mass, where the same force would have a smaller impact.


 * Sample Calculations**



In this lab, we studied collisions that applied to the Law of Conservation of Momentum. From this law, we were able to hypothesize two things. First, we predicted that the initial and final momentum of our system would be equal, since the law states that momentum is conserved. Second, we hypothesized that elastic collisions would be the ones that conserved kinetic energy and inelastic collisions would not. To test out these hypotheses, the class was split up into different groups that each focused on a different type of collision. Our group, however, simulated an "explosion" collision. To examine this type of collision, we used two carts and a plunger that "exploded" them away from each other. The, we used the motion sensors to determine the velocities of each cart after the explosion. After doing this same simulation with five different masses, we were able to come to the conclusion that our first hypothesis, stating that the initial momentum was equal to the final momentum, was correct. Throughout this experiment, final momentum was calculated as being nearly equal to initial momentum, demonstrating how it was completely conserved throughout our entire lab. For most groups, they were able to use percent difference to determine the accuracy of their results. However, our group was unique because if we were to use percent difference, we would obtain a result of 200% exactly each time. To fix this, we used a different method of determining the validity of our experimental data. Our percent difference was simply the absolute value of the difference between our initial and final calculated momentums. This allowed us to see how close the final and initial momentums were to each other, since if we used another method our percent difference would continuously come out to be 200% exactly. After determining our percent differences, we were able to conclude that we had extremely accurate results. For example, all of our percent differences ranged from .00051-.00096. One can easily see that all of these numbers are very small, therefore showing that our experimental data was extremely accurate. Next, we also tested to see if the "explosion" was an elastic or inelastic collision. From analyzing our kinetic energy values, we were able to see that this was an inelastic situation, since none of this energy was conserved. Our situation always started at rest, and ended in motion. Thus, one can see how no kinetic energy was conserved throughout this process. Although our results proved to be successful, there were numerous sources of error in this experiment. First, the Law of Conservation of Momentum states that the objects must be isolated in these situations. However, the system in this situation (the two carts) was not completely isolated, as they were sitting on a track that evidently created friction throughout the lab. This caused a loss of energy to friction that could have affected our results. To fix this, we could have attempted to complete the lab in a more isolated environment, where friction would not be a contributor to the loss of energy. Another source of error could have come from a simple misreading of the graphs. Often, the graphs did not look exactly how we thought that we would. Because of this, we often had to estimate as to which point was the initial velocity and which point was the final velocity. To fix this, we could have set different intervals on the graph from the beginning, therefore allowing us to have a clearer view of the data. This lab also applies to a great deal of real-life situations. For example, if a meteor is in space, it might split into two smaller pieces, therefore categorizing it as an explosion.
 * Conclusion**

=**Ballistic Pendulum Lab**=

__Part A:__ Jake Aronson __Part B:__ Dani Rubenstein __Part C:__ Jess Smith __Part D:__ Michael Solimano

What is the initial speed of the ball fired into a ballistic pendulum?
 * Objective**

We hypothesize that the initial speed of the ball will be determined by three methods: through using a photogate, through projectile kinematics, and through the Law of Conservation of Momentum (Ballistic Pendulum) / the Law of Conservation of Energy. We predict that the photogate method will be the most successful because it requires few calculations and basically comes directly from Data Studio.
 * Hypothesis**

During this lab, we had to go through three different procedures to test the three different methods of finding initial velocity. For each method, the steel ball is pushed into the launcher at a medium range at a 0 degree angle. The first method involved using a photogate. The photogate was held right where the ball would be released and the USB cord was plugged into the computer, allowing us to access Data Studio. We then were able to use this to record the time in gate, thus allowing us to calculate the initial velocity. However, to do this, we had to measure the diameter of the ball with a meter stick to obtain our results. This method was performed 5 times and an average was calculated using Microsoft Excel. The next method involved kinematics and projectiles. First, we had to use a meter-stick to measure the vertical height of the projectile. Next, we placed carbon paper on the ground in order to record the horizontal distance of the projectile. Using a measuring tape, we were able to measure the different horizontal distances after launching the projectile five times. From the average of these results we were able to fill out the x and y columns of our chart and solve for the initial velocity. The final method measured velocity using the Law of Conservation of Momentum. The ball was launched into a pendulum that hung from the launcher. We then were able to use this to measure theta, which provided us with the height because we were then able to use L-Lcostheta. Finally, we created a Data chart to organize all of the methods and analyzed and reviewed our results through Percent Difference.
 * Methods and Materials**

media type="file" key="projectile.mov" width="300" height="300"
 * Video of Projectile Method**

media type="file" key="photogate.mov" width="300" height="300"
 * Video of Photogate Method**

media type="file" key="ballistic pendulum.mov" width="300" height="300"
 * Video of Ballistic Pendulum Method**


 * Data**




 * Sample Calculations**

1. In general, what kind of collision conserves kinetic energy? What kind doesn’t? What kind results in maximum loss of kinetic energy?
 * Analysis/Discussion Questions**
 * Elastic collisions conserve kinetic energy while inelastic collisions do not. When two objects are moving with the same momentum and have a head on collision where they stop at rest, this yields the greatest loss in kinetic energy.

2. Consider the collision between the ball and pendulum. a.Is it elastic or inelastic? b.Is energy conserved?
 * It is an inelastic collision.
 * Energy is not conserved because in an inelastic collision the energy is not conversed.
 * 1) Is momentum conserved?
 * Momentum is conserved because the law of conservation of momentum can be applied.

3. Consider the swing and rise of the pendulum and embedded ball. a.Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height?
 * No, the energy is not conserved right before the ball strikes the pendulum because it is an inelastic collision. The ball loses kinetic energy therefore, the energy is not conversed.

b.How about momentum?
 * The momentum is conversed due to the Law of Conservation of Momentum.

4. It would greatly simplify the calculations if kinetic energy were conserved in the collision between ball and pendulum. a.Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum. b.What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy. d.Calculate the ratio M/(m+M). Compare this ratio with the 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. These numbers are almost exactly the same which shows that our data collection was good.
 * 1) According to your calculations, would it be valid to assume that energy was conserved in that collision?
 * 2) No, the energy was not conserved because the percent lost is very high.

5. Go to [] 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.)
 * 1) When the mass of the ball is increased, the initial velocity increases as a result, which makes theta and the height also increase. When the mass of the pendulum is increased decreases both theta and the height.

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 momentum method gave better results?
 * The velocities of the kinematics method and the photogate method are very similar. However, the velocities of the Ballistic Pendulum method were much higher than the other two methods. This could be due to inconsistencies with the launcher, poor measurements, or misreadings of the photogate timer. If we were to build our own Balistic Pendulum, we would have a digital reading of the angle so we did not have to adjust it ourselves each time the ball was launched. Also, we would make it possible to use a photogate the calculate the initial velocity before the ball entered the pendulum.

In this lab we sought to analyze a ballistic pendulum, a system where a ball is fired into a catcher, which subsequently moves based on the ball hitting it. In this lab we used our knowledge of the Law of Conservation of Energy, projectiles, and kinematics to analyze the speed of the ball leaving the projectile launcher, and its subsequent affect on its “catcher”. In doing this, we sought to find the initial speed of the ball. To do this, we used a photogate to measure its time in the photogate, which was used to solve for the initial velocity, using a simple velocity equation. We also fired the projectile onto carbon paper, and analyzed it’s average distance in order to use projectile motion techniques to solve for the initial velocity. Finally, we employed our knowledge of the Law of Conservation of Energy to solve for the initial velocity. These values were 5.44 m/s, 5.71 m/s, and 1.12 m/s respectively, with the highest average percent difference of these being .88. Ideally, these values should all be equal, as there was in fact one initially velocity with which the ball left the launcher. We also sought to find the height that the pendulum catcher rose after being struck by the ball. In order to find this, we found the angle that the pendulum created, and found an average of many trials of this. We then used our known relationship of L – L cos(theta) = h, to solve for the experimental height that the pendulum rose to. This height value was then used in a Law of Conservation of Energy equation to again solve for experimental initial velocity. These values all were necessary in completely analyzing a projectile launched into a catcher.
 * Conclusion**

There were sources of error in this lab that could have contributed to our percent differences. One source of error was found in the pendulum. The pendulum was not clamped to the table, and was thus free to move when firing. This could have led to differing x-distances for our projectile motion, which could have further contributed to error in the values we obtained. Also, it was not easy to ensure that the photogate timer was at the exact exit of the shooter, and thus would not be measuring the //initial// velocity. This could have further contributed to error when solving for our average initial velocity. These differing values would have led to error in our initial velocity calculations, and thus in our analyzing of the ballistic pendulum.

Had we conducted this lab again, we could have used a means of clamping the launcher to the table. This could have reduced the error in our lab, and improved our projectile motion experimental results. Also, we could have used a better means of assuring that our photogate timer was in fact at the exit of the shooter. By doing this, we could have reduced the chances of error and its implications stated above. This lab also did have actual implications. Many theme park rides follow this model, and these concepts would be important to make sure that the ride is safe, and in analyzing the ride itself.