Group2_2_ch6

= = toc =Lab: Law of Conservation Energy 2/1= Task A: Ali Task B: Sammy Task C: Danielle Task D: Ben

Station 1: What is the speed of the cart at the bottom of the incline? Station 2: What is the speed of the ball when it leaves the launcher? Before it hits the ground? Station 3: What is the speed of the mass at the lowest point of the swing? Station 4: Where is the max height of a ball at the top of an incline? Station 5: What is the speed of the ball as it leaves the launcher? Station 6: What is the speed of the "roller coaster" at the top of the loop?
 * Objective**: What is the relationship between changes in kinetic energy and changes in gravitational potential energy?


 * Hypothesis**: Due to the law of conservation of energy, which says that energy cannot be created or destroyed, the initial energy of the object should equal the final energy of the object. This idea has been described to the class previously in both homework and in class and should play a role in this lab.

Station 1: We put the picket fence piece on the cart. We let the cart roll down the metal ramp. Although the car didnt pass through the photogate, the picket fence did so we used that to calculate velocity. We measured the intial height (top of ramp) and final height (photogate) with a meter stick.
 * Methods and Materials:**

Station 2: We launched a ball at short range through a photogate. We measured the diameter of the ball, distance between photogates and recorded the time through the initial photogate and final photogate to get velocity.

Station 3: We measured the initial height to be 20 cm above the bottom of the wooden cylinder and the bottom of it to be the final height. We recorded the time it was in a photogate and the diameter of the cylinder to find velocity.

Station 4: We measured the height of the shorter incline and the point on the longer incline to which the ball went up to with a meter stick. We know both the initial and final velocity are 0.

Station 5: We launched the ball through the vertical launcher and recorded the time it was in the photogate. We measured the diamter of the ball.

Station 6: We measured the initial height and the height at the top of the loop with a meter stick. We dropped the ball down the ramp and recorded the time the ball passed through the photogate at the top of the loop.

Data and Observations: Sample Calculations: Analysis:





Conclusion: Based on the results of this experiment, our hypothesis that the initial energy and the final energy must be the same is correct. This was shown in our lab because the experiments at which we calculated similar initial and final energies had the least percent difference. As a result, these experiments were the ones with the least error.

For each station, the percent difference was calculated to measure the amount of error in each experiment. The percent difference in each station, the difference of initial and final energy over the average of the two, was calculated as: 9.11%, 77.31%, 67.59%, 11.58%, 3.13%, and 35.63% (this is in numerical order). A possible source of error could have been the exclusion of friction in the determinations. Friction is a kind of force that performs work on a moving object; work can cause an object to either take in or give off energy and therefore is usually included in the equation of the law of conservation of energy. In our calculations, work caused by friction was not included, leading to differences in the amount of initial energy and the amount of final energy calculated for each scenario. To erase this error, we could have done one of two things: either find a way to calculate friction and thus calculate the work at each station or perform the experiments in a frictionless environment. Doing this would decrease the difference between initial and final energy and thus would decrease percent difference. At stations 2 and 3, high percent differences were calculated. At station 2, the high percent difference may have come from imprecise measurements of initial height. At station 3, the high percent difference may have come from imprecise measurements of diameter of the pendulum. To decrease the error at these stations, we could have used a more precise meter stick that would have given more accurate measurements.

As implied from the experiment, the law of conservation of energy can be applied to a multitude of situations, many of which can be found in real life. For example, the logic behind station 1 could be applied to a car rolling down a hill. A similar situation to station 6 exists when a roller coaster goes to the top of the inside of a loop. Like in the labs, initial and final energies can be calculated based on the various factors in each of these situations.

=Lab: The LCE for a Mass on the Spring 2/7= Task A: Danielle Task B: Ali Task C: Ben Task D: Sammy

**Objectives:**
 * 1) Determine the spring constant k of several springs by measuring the elongation of the spring for specific applied forces.
 * 2) What is the elastic potential energy of the spring?
 * 3) What is the graph of the stretched spring and how much work was done?
 * 4) Measure the gravitational potential and kinetic energy at 3 positions during the spring oscillation.

Using the given information from the "spring box", we hypothesize that the red spring, since it is the softest, will have the smallest k value, while the green spring, which seems to be the "strongest" spring, will have the largest k value. The Elastic Potential Energy can be determined by making the hanging mass and the spring have equilibrium. When they are in equilibrium, both forces must be equal. This can be accomplished by finding a mass (or adding to a mass), so that the spring stays still and does not oscillate. We can then graph these data points and take the slope of the line to find the spring constant k. The GPE and KE can be found at three positions during the spring oscillation: "equilibrium", maximum, and minimum. The minimum displacement is the lowest point where the spring is pulled to. The maximum displacement is the point where the spring goes the highest after being released. By finding their position and velocities, we can determine the GPE and KE of the mass.
 * Hypothesis**:

We will be using springs that are attached by a rod to a clamp and various masses to find EPE. We will use a balance scale to measure the masses and a meter stick to measure the distance the spring stretched out.
 * Methods and Materials**:

For the second part of the lab, we used a .5 kg mass attached to the spring with cardboard tape to the bottom to maximize surface area. The total mass was about .559 kg. Therefore, the motion sensor directly underneath would easily detect any movement of the mass. A picture of the materials used in Part 2 is below:

Free body diagram for Part A:
 * Data and Observations**:





Part B: [[image:data_for_data_studio_[art_b_group_7.png]][[image:Screen_shot_2012-02-08_at_9.44.41_AM.png]] __Part A__ Percent Error:
 * Sample Calculations/Analysis**:

Red: Blue: Green: White:

Percent Difference: Red Blue

Green

White

__Part B__ Percent Difference: Minimum: Equilibrium: Maximum:

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? <span style="font-family: Arial,Helvetica,sans-serif;">The data for the displacement of the string vs. the applied force DOES show that the spring constant will be constant for a particular range of forces. This is because the graph for each string is linear; therefore, each graph has directly proportional force and displacement, meaning that each graph has a constant slope. This in turn means that each string has a constant string constant (which is reflected in the slope).
 * Discussion Questions:**

<span style="font-family: Arial,Helvetica,sans-serif;">2. How can you tell which spring is softer by merely looking at the graph? <span style="font-family: Arial,Helvetica,sans-serif;">The spring with the smallest spring constant will be the softest string. Therefore, to determine which string is softest, one must determine which of the linear graphs has the smallest slope. In this situation, the softest string is the red string. Additionally, by looking at our graph, you can tell which one has is softest by seeing the displacement. The one with the greatest displacement most likely has the "softest" spring constant.

<span style="font-family: Arial,Helvetica,sans-serif;">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: Arial,Helvetica,sans-serif;"> When the total mass is at the "minimum" Elastic Potential Energy and Gravitational Potential Energy are present. This is because the velocity is close to zero. At equilibrium, there is Kinetic Energy, Gravitational Potential Energy, and Elastic Potential Energy present. At the "maximum", there is solely Gravitational Potential Energy. Like the minimum, there is zero velocity. However, it is important to recall that at ALL points in the trial, there is the same amount of energy. This conclusion is derived from the Law of Conservation of Energy.

Our results for Part A of this lab we very wide ranged. We were correct in our hypothesis in that the red spring will have the highest K value (slope of the line) because it is the softest. Even though the box predicts that the white will have the highest K, we predicted that the green one will and that's what we got it. This can be for a number of reasons: 1) we didn't take the white spring into account or 2) it is taken account into the percent error. For red and blue, we obtained results under the 10% error mark; however, for the green and white spring our percent error was above that mark. A number of sources of error were present here. One in particular though, could be the number of spring that are on the holding device at once. If there are three cords with weights on one side of the device compared to if the device has a balanced number of strings can affect the results. If the device is unbalanced in terms of weight, then the weight component of the hanging mass may have components rather than pulling straight down. The only way to counteract this source of error is to put the same cords and weight on each side will measuring. As for Part B, we had relatively low percent errors in 0.934%, 5.6%, and 6.6%. For this part, one possible source of error could be in the measurments. We used meter sticks for the initial mesurements which are not very precise. For both of these parts a real life could be bun-jee jumping. Just like this lab, you need to take hanging weight and K into account for both velocity and distance stretched.
 * Conclusion**:

=Roller Coaster Project= We proudly introduce to you... COMING SOON TO A THEME PARK NEAR YOU!!!!

__Blueprint of Burnin' Bullet__ Side View:

Top View:

__Pictures of Prototype__ Side View: Top View:

Our roller coaster begins three columns up, and has a long initial drop to gain speed needed to enter the loop. After its first initial drop, the marble goes up the first hill. Then, to gain speed, the ball descends from this hill and enters the vertical loop. Unfortunately, even with all of the extra drops we added, our marble still was not able to gain enough speed to enter the vertical loop. If the marble was successful, which was only a couple times out of hundred, then it would enter a wide turn, which would lead it to the horizontal drop. Once the marble goes through the horizontal loop, it goes up our second hill and then goes down a slight drop, which takes it to the end of our roller coaster.

__Data and Calculations__ Experimental: Theoretical: Percent Error: A) B) The //theoretical// amount of energy and power required to get the roller coast rolling. (time=30 seconds)
 * At some instances, percent error is extremely high because the ball had to be moved without going through a previous section (up to the vertical loop). As a result, the actual velocity was different than it would have been if it had gone successfully through the loop.

C)

D) Minimum Height Requirement For Amount of Total Energy This height is the requirement that needs to be higher than the loop. Therefore, the ACTUAL minimum height should be .443 meters.

E)

F)

__Discussion of Roller Coaster__ Overview: Burnin' Bullet was made with two elements in mind: it had to have the various common features of roller coasters in many amusement parks and it had to be fun for any riders who might ride it. We started off with a hill with which the ball was supposed to accelerate (or increase its velocity over time) enough, so that the ball would move at high enough velocities to move through the entire roller coaster. After a small roll upward, the ball then went down a second hill that then transitioned into a vertical loop. After this, the ball went through a curve and then through a horizontal loop that wrapped around a support. From here, the ball went through a straight path and then went down a final hill (the end of which was the end of the roller coaster).

Acceleration/Apparent Weight/Circular Motion: The key to this prototype was acceleration. To find the acceleration along the straight paths, we found the velocity (by using the photogates) at the two endpoints of the path. With this information, we have "the change in velocity / the change in time", which is what acceleration is equal too. As for cases where the ball was moving with circular motion and had centripetal acceleration, the ball required a minimum velocity (which could only be achieved by acceleration) to make it through these portions (like the vertical loop). In the case of the vertical loop, the ball needed a minimum velocity so that its apparent weight (the normal force of the roller coaster on the ball) would not bring it down. This minimum velocity was found through the equation provided by Newton's Second Law (net force = mass x acceleration, acceleration being velocity^2/radius). As one can see, we needed to use circular motion equations to figure out the minimum speed. Apparent weight is the "weight" of the object at that position, where gravity and Normal force are the only two acting forces. At different points of the roller coaster, the object may "feel" as if it is a different weight. For example, at the top of hill, the sensation of weightlessness may be felt as normal force and gravity are pointing in different directions. At the bottom of a hill, the object may feel at its heaviest as normal force and gravity are pointing in the same direction. By using apparent weight, we were able to solve minimum speed, as shown above.

Newton's Laws: As shown in the equation above in Part C, we found the theoretical minimum speed of .89 m/s at the top of the vertical loop. From our data in the chart above, we found that the actual speed was 1.040 m/s. By using Newton's Second Law (F=ma), and applying our knowledge of circular motion laws, we were able to find that the acceleration at this point was 33.766 m/s/s. We had previously learned that the maximum amount of acceleration that a human can tolerate is approximately 4 "gs", which is equal to (4)(9.8)= 39.2 m/s/s. Therefore, our coaster had passed the "safety requirement" at the vertical loop as this acceleration was less than the max. The only area at which acceleration exceeded 4 gs was the second hill at which the ball accelerated 42.99 m/s (which is equivalent to 4.39 gs). Correcting this would only require either an increase in the angle of the incline or a decrease in the vertical distance traveled by the ball (or a combination of both).

The other Newton's Laws can also be applied to this roller coaster. According to the First Law, an object will continue in its state of motion, unless acted on by an outside force. This was true of the ball that was on our roller coaster which was acted on by forces like gravity, normal force, and friction, allowing it to move through various parts of the roller coaster. For example, when going down a hill, the ball is acted on by gravity, normal force, and friction to get it down from the top of the hill. If this didn't happen, the ball would roll at a constant rate in the same direction. The Third Law (for every action, there is an equal and opposite reaction) can apply to this roller coaster because (if the ball were to be replaced by an actual cart with people in it) the normal force exerted by the seat of the cart on the riders would be equaled by an opposing normal force by the people on the cart. In places like the vertical loop, this would ideally keep people from being ejected from their seats.

Energy Conservation: Theoretically, our coaster was supposed to conserve energy, as stated by the Law of Conservation. In a perfect scenario, there would be no forces (like friction) doing work on the ball and no energy would be lost in the form of heat. However, the paper from which the roller coaster was built was not a frictionless surface. As a result, work WAS done on the moving ball, causing an expenditure of energy in the form of heat. Work was also being done by air resistance on the ball. As shown by the sample calculations, by the end of the third hill, there was nearly 0 J of total energy left. According to the work equation, work is equal to Force*distance*cos(theta). Because the work being done on the ball (friction) was acting in a different direction than the ball's movement, it meant theta was equal to 90 degrees. The cos(90) is equal to -1, meaning work is negative, showing why the ball lost energy and didn't gain any.

Power: For Part B, we found the power to get the marble to the initial drop. Power is equal to "work/time". Because this initial incline is simply hypothetical and we didn't actual build an engine to get the marble to the initial drop, we were able to decide what the time would be and chose 30 seconds. According to the Law of Conservation of Energy, in this situation from the ground to the initial drop, work is equal to gravitational potential energy which is equal to "mass*gravity*height of initial drop". We took the mass of the marble, the measure of gravity (9.8 m/s2), the height of the initial drop, and our allotted time of 30 seconds to find the required power to get the marble up. The units of power can be horsepower or watts (746 W = 1 hp); we chose to measure the power in this situation in watts.

Spring System/Hooke's Law: <span style="background-color: #ffffff; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">For Part F, we had to find the distance that a potential spring system to stop the roller coaster at the end of the third hill should be, in case the brakes that will be put in on the real model fail. In order to calculate the elastic potential energy for this spring system, Hooke's Law had to be used. The equation for Hooke's Law: EPE=k*x, where the extension of the spring is directly proportional to a force. However, for us, we decide to set EPE equal to the energy dissipated at the bottom of the hill. First, we needed to find the minimum stopping distance, to ensure that there would have a great enough distance for the cart to travel and stop safely. We found the minimum distance to be .73 meters, and thereafter chose a stopping distance of .8 meters. We then plugged this distance in for x, and solved for k. Our spring constant k came out to be .016 N/m.

Uncertainties/Sources of Error: There were various inherent uncertainties and sources of error in crafting the prototype of this roller coaster, which had an impact on our calculations. First and foremost, on many occasions the ball was unable to make it past the vertical loop, despite the fact that it exceeded the required minimum velocity. This could have come from the fact that the vertical loop (as well as the rest of the prototype) was made out of flimsy paper that weakened over time. Especially with the fact that the ball was metallic in mind, it is possible that the weakness of the material may have hindered the ball's movement (we noticed that the ball was somehow banging against the loop, preventing it from going up). As a result of the ball's inability to make it through the loop, we were forced to find move the ball through certain portions of the roller coaster without letting it go through the beginning stages. Ultimately, this resulted in different velocities than we would have gotten otherwise, leading to greater percent error. Other uncertainties came from the fact that neither friction nor air resistance were taken into account in our theoretical equations; therefore, the theoretical and actual velocities were different. At the same time, another potential source of error could have been through our measurements of height. For this, we used a meter stick; however, because hundredths were required when measuring height, we were forced to estimate. This may have led to theoretical velocities that might have led to greater percent error. To fix all these sources of error, we would have to utilize a sturdier material to make the roller coaster (like cardboard) to rebuild the vertical loop, take friction and air resistance into account when using the Work-Energy Theorem, and use a ruler that could measure in hundredths of meters.

Conclusion: Even though the ball was unable to make it through the prototype on most tries, the acceleration at the second hill exceeded 4 gs, and there was large percent error at some points, the fundamentals of our roller coaster are sound. All the required parts of this kind of roller coaster are included and we were able to calculate theoretical values, find actual values, and calculate percent error. If given more time, we would fix specific errors of our prototype (like improving the vertical loop and decreasing the amount of acceleration when going down the second hill) and make it so that the ball would be able to make it through the roller coaster numerous consecutive times without an issue. Therefore, consider this a prototype that is the first step towards creating the perfect roller coaster for your theme park! Once all the necessary corrections are made, Burnin' Bullet can be constructed, opened to the public, and will be safe for all to ride. Look out for Burnin' Bullet in the summer of 2012!!!

__Videos__ media type="file" key="Danielle Bonnett, Ali Cantor, Sammy Caspert, Ben Weiss Group 7 .mov" width="300" height="300" media type="file" key="Movie on 2012-02-27 at 21.18.mov" width="300" height="300"
 * Presentation Day Video**
 * Video Of Marble Ball Success**


 * After the Presentation Day, we recorded a video of a marble successfully going through the prototype. This demonstrates that with better conditions (a lighter ball to match the somewhat flimsy paper), the prototype was able to work and that Burnin' Bullet can eventually become a reality.

=Elastic and Inelastic Collisions Lab= A: Ben Weiss B: Ali Cantor C: Danielle Bonnet D: Sammy Caspert


 * Objective**: 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?

If the relationship between the initial momentum and final momentum is determined, one will find that the total initial momentum will equal the total final momentum (this comes from the Law of Conservation of Momentum). If different kinds of collisions are compared, the ones in which kinetic energy is conserved will be elastic collisions and the ones in which kinetic energy is not conserved will be inelastic collisions (this comes from our previous class discussions of the difference between elastic and inelastic collisions).
 * Hypotheses**:

In order to determine the relationship between initial and final momentum, the class had to set up scenarios in which momentum of two different moving objects could be calculated. Specifically, the class used two carts (Cart A and Cart B) that were rolled on a straight ramp in a variety of different ways.
 * Methods and Materials**:

The following situations were observed: A colliding with B at rest, A and B moving towards each other (plunger out), an explosion causing both A and B to move from rest, A and B moving in the same direction and eventually bouncing off each other, and A and B moving in the same direction and sticking via velcro. For each scenario, 5 different tests were performed using varied masses. The masses of the carts were increased by adding several metallic weights onto them (the removal of which decreased the masses). Our group observed A and B moving towards each other with the plunger out.

Next, the initial and final velocities of each cart were measured on Data Studio using graphs created from our method. To create these graphs, we had to create the required collision and measure velocity through sensors on either side of the track. After doing so, we were able to take the Law of Conservation of Momentum (m1v1i +m2v2i = m1v1f + m2v2f) to determine whether or not momentum was conserved. If the total initial momentum and total final momentum were the same (or extremely close), then our hypothesis that momentum was conserved would be correct. Next, we had to use the Law of Conservation of Energy (TEi = TEf) to determine whether or not our particular scenario was an elastic collision (in which kinetic energy was conserved) or an inelastic collision (in which kinetic energy was not conserved). If initial kinetic energy and final kinetic energy were equal or extremely close, then our collision would have been elastic. If not, it would have been inelastic.

Below are a picture of the materials used and a recording of our procedure: media type="file" key="Movie on 2012-03-14 at 08.32


 * Data and Observations**:



Actual Total KE (Using Cart A, Initial Velocity, Cart B, Initial Velocity) These calculations of initial and final momentum demonstrate how momentum is conserved during this collision (the slight difference in values can be attributed to experimental error).
 * Sample Calculations/Analysis**:


 * Analysis Questions**:
 * 1) Is momentum conserved in this experiment? Explain, using actual data from the lab.
 * 2) It appears that momentum is conserved in this experiment. All of the values for initial and final momentum are VERY close. Take the first situation, for example. The initial momentum= .0226, while final momentum =.02375. These values have a percent difference of 2%, which is very small. Therefore, we can conclude that, if we put aside human error and other forms of work, that momentum was conserved.
 * 3) When carts of unequal masses push away from each other, which cart has a higher velocity? Explain why this is.
 * 4) The cart with the smaller mass has a higher velocity. This is because the force (from the cart with more mass and some velocity) exerts a greater force on the smaller-massed cart, than the smaller-massed cart exerts on the big cart.
 * 5) When carts of unequal masses push away from each other, which cart has more momentum?
 * 6) If carts of unequal masses push away from each other, they have equal momentum. Since, mass • ∆velocity=p, as shown above, each result has equal value, as shown in the "explosive collision."
 * 7) Is the momentum dependent on which cart has its plunger cocked? Explain why or why.
 * 8) No, the momentum is independent of which cart has the plunger out. The momentum equation, p=m(changein)v, shows that the momentums for each of the carts are very similar. It is because, as the mass of one of the carts increases, its velocity decreases. In most cases, the velocities are relatively the same. Additionally, since Newton's 3rd Law states that every action has an equal and opposite reaction, we can conclude that the force on both of the carts is the same.

According to the Law of Conservation of Momentum, we hypothesized that the initial and final momentum of an isolated collision must be equal. At the same time, we also tested to see whether a specific collision was elastic or inelastic. We hypothesized that an elastic collision, considered to be perfect, is one in which no kinetic energy is lost. We also hypothesized that in an inelastic collision, there is a difference between the initial kinetic energy and final kinetic energy. This lab was unique in that there were several scenarios which had to be examined, yet each lab group in the class was assigned one specific scenario. We simulated a situation in which two objects collided and bounced apart due to a spring at the end of one of the carts. After running five trial with varied masses, we found that our results matched our first hypothesis that momentum would be conserved. Throughout the experiment, final momentum was calculated as being nearly equal to initial velocity, demonstrating how momentum was being maintained (for the most part) throughout the entire scenario. The validity of our results is demonstrated by our calculations of percent difference (the absolute value of the difference between final and initial momentum over the average momentum) for our five trials. These values range from 1% to 17%, very low values that show that our results were extremely accurate. For the most part, the other groups also had low percent differences (a trend was that collisions that were found to be inelastic had higher percent difference). The only collision with unusual high percent difference was the explosion between the two carts, which was found to consistently have 200% difference. However, this would make sense because the final velocity of the carts was always much higher than the initial velocity for both (0 m/s). Percent difference is not a perfect evaluation for explosions, but is necessary to maintain a single method of calculating error for all the collisions.
 * Conclusion:**

At the same time, we also this situation to be an inelastic collision. Through calculations, we observed that kinetic energy was not conserved throughout our scenario. Based on our previous knowledge from class, we hypothesized that if kinetic energy were conserved throughout the collision, the collision would be perfectly elastic. Because it was not, we reasoned that it was an inelastic collision, following the reasoning behind our hypothesis. Based on this given information, we determined that collisions 2,3, and 4 were inelastic, while collisions 1 and 5 were elastic.

There were a number of sources of error in this experiment. First, according to the Law of Conservation of Momentum, the collision must be isolated, meaning there is no surrounding environment. In this lab however, the collision was not completely isolated. The carts moved along a track, through which energy was lost to friction, meaning that the velocities decreased to amounts that they would not ideally be. Another source of error could be from the use of our hands to give the carts an initial push. Because we did this, our hands may have gotten in front of the sensors, which would have affected the results of the lab. If we could have moved the sensors or our hands to avoid each other, this could have prevented this error. A third source of error could have come from the reading of the graph. The graphs never came out ideally (which we assumed would happen), so occasionally we would have to estimate or choose a specific point ourselves for the initial/final velocity of the crash. To decrease the amount of error, we could have performed this lab in a more isolated environment, positioned the sensors so that our hands would not interfere with the data collection, and use graphs with easier to read intervals.

=Ballistic Pendulum Lab 3/21= **Objective** What is the initial speed of a ball fired into a ballistic pendulum?

**Hypothesis** Despite the three different methods (ballistic pendulum method, photo gate timer, and kinematics), the initial speed of the ball should remain relatively the same. Also, we hypothesize if the initial speed is found, it will be found to be about 4.70 m/s. This is based on the results of the "Ball in a Cup" Lab that was done in Chapter 3.

<span style="font-family: Arial,Helvetica,sans-serif;">First, we found initial velocity by firing the metal ball into a ballistic pendulum. We attached a ballistic pendulum to a typical launcher setup. Then, we launched the ball at medium range into the ballistic pendulum, where we thereafter recorded the angle (** θ). ** Then, we set the angle indicator to angle 1-2 degrees less than which reached before. We repeated this procedure five times, and recorded the maximum angle reached by the ballistic pendulum. We found that each time the angles were all very similar. We then removed the ballistic pendulum from the launcher and proceeded to use the bare launcher, along with a photo gate timer, to measure the speed. Again, we repeated this method five times, and recorded each time. We then used the distance of .025 of the ball, divided by the time in photo gate to get the speed of the ball. Lastly, we found the initial velocity of the ball by using kinematics. We launched the ball at medium range from the launcher onto carbon paper on the ground. We then used measuring tape to measure the distance traveled by the ball.
 * Methods and Materials **

Ballistic Pendulum: media type="file" key="Movie on 2012-03-21 at 08.30.mov" width="300" height="300"

Photogate: media type="file" key="Movie on 2012-03-21 at 08.53.mov" width="300" height="300"

Kinematics: media type="file" key="Movie on 2012-03-21 at 08.45.mov" width="300" height="300"

**Sample** **Calculations**
 * Data and Observations **

Velocity Using Kinematics:

Velocity Using Photogate:

Ballistic Pendulum: Percent Difference Sample:

<span style="font-family: Arial,Helvetica,sans-serif;">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 will conserve kinetic energy, while inelastic collisions will not. A collision that is perfectly inelastic will have a maximum loss of kinetic energy.
 * Analysis Questions **

2. Consider the collision between the ball and pendulum
 * 1) Is it elastic or inelastic? The collision is inelastic.
 * 2) Is energy conserved? Energy is not conserved in this collision because it is inelastic.
 * 3) <span style="font-family: Arial,Helvetica,sans-serif;">Is momentum conserved? Momentum is conserved in this collision because, like all collisions, it follows the Law of Conservation of Momentum.

<span style="font-family: Arial,Helvetica,sans-serif;">3. Consider the swing and rise of the pendulum and embedded ball.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif;">Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height? No, energy is not conserved because the collision is inelastic. Ultimately, the ball loses its kinetic energy as it collides into the pendulum.
 * 2) <span style="font-family: Arial,Helvetica,sans-serif;">How about momentum? Momentum is conserved because all collisions follow the Law of Conservation of Momentum.

<span style="font-family: Arial,Helvetica,sans-serif;">4. It would greatly simplify the calculations if kinetic energy were conserved in the collision between ball and pendulum. <span style="font-family: Arial,Helvetica,sans-serif;">
 * 1) <span style="font-family: Arial,Helvetica,sans-serif;">Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum.
 * 2) <span style="font-family: Arial,Helvetica,sans-serif;">[[image:Screen_shot_2012-03-22_at_11.01.26_AM.png]]
 * 3) <span style="font-family: Arial,Helvetica,sans-serif;">What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy
 * 4) <span style="font-family: Arial,Helvetica,sans-serif;">[[image:Screen_shot_2012-03-22_at_11.00.49_AM.png]]
 * 5) <span style="font-family: Arial,Helvetica,sans-serif;">According to your calculations, would it be valid to assume that energy was conserved in that collision? No it was not conserved, as you can see we got a very large percent lost of 78.52%. Therefore, we cannot assume it was conserved and must conclude that this was an inelastic collision.
 * 6) <span style="font-family: Arial,Helvetica,sans-serif;">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. When multiplied by 100, this is very comparable to our percentage lost value of 78.52. There is a .1146% difference between the two.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif;">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. When multiplied by 100, this is very comparable to our percentage lost value of 78.52. There is a .1146% difference between the two.

<span style="font-family: Arial,Helvetica,sans-serif;">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.) The increase of mass of the ball accounts for a higher height that the entire pendulum will reach. The increase of mass of the pendulum bob accounts for a decrease in height that the entire pendulum will reach. In conclusion, it appears that the larger the mass of the ball the higher the height the pendulum reaches, while the larger the mass of the pendulum the lower the heigh the pendulum reaches.

<span style="font-family: Arial,Helvetica,sans-serif;">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? No, there is not a significant difference between the three calculated values of velocity. However, each of the methods had their sources of errors. When using the projectile launcher for kinematics, we had to measure the distance that the ball traveled. There may have been error in our measurements as we had to use some estimations. Although the photogate was extremely accurate, and by far the best method, there too could have been some error due to friction and the measurement of the diameter of the ball. Both of these factors would have effected the calculated value of velocity. In regards to the ballistic pendulum, we did our best to prevent friction from becoming a major factor in our results. One idea that future physicists could use is to make the launcher have a sensor, like the photogate, that can detect the angle that the ball and pendulum reach. Also, if the ball and catcher, as they rose in the air, followed some sort of track, similar to that of a rollercoaster, it would reduce error. This would prevent the ball and catcher from shaking and losing energy to work.

Prior to our lab, we hypothesized that for each of the three different methods required to find initial velocity, the calculated initial velocities would all be relatively the same. Through our results, this hypothesis was proven true: we calculated three different values that had small percent differences between them (3.404 m/s, 3.63 m/s, and 3.49 m/s). 3.404 m/s was found through kinematics. Similar to methods employed in the Ball in a Cup Lab, we launched the metal ball from the launcher (which was clamped onto the table), so that it would land on a piece of carbon paper several times. From here, we measured the vertical distance from the table to the ground and the horizontal distances from the launcher to the marks left by the ball's landing (which were then averaged). After this, we used a kinematics equation for vertical distance to find the total time traveled, which we then plugged into a kinematics equation for horizontal distance to find the initial velocity. The second method (through which we found 3.63 m/s) was through the use of a photo timer. This was a very simple method in which we divided the distance the ball traveled as it went through the photo gate (its diameter) and then divided it by the average time it took to get through. The third method (through which we found 3.49 m/s) was through a ballistic pendulum. After finding the angle the pendulum moved when the ball was shot into it, we used the Law of Conservation of Energy, then the Law of Conservation of Momentum to find the initial velocity of the launcher. As a whole, our results were very consistent and did not have much error. The percent difference for the kinematics method, the photogate method, and the ballistic pendulum method (when compared to the average of the three) were 2.53%, 2.56%, and .94% respectively.
 * Conclusion **

Looking at this data, one can observe that our other hypothesis was wrong regarding the initial velocity of the ball. Instead of being 4.70 m/s, the average initial velocity was instead 3.523 m/s. There is a simple reason for this; the ball that was used for this experiment was different than the one used in the Ball in a Cup Lab. Unlike in that lab, this lab required a metallic ball with a greater mass. Therefore, it would make sense that the velocity was less than for the lighter plastic ball that was used in Ball in a Cup.

There are several sources of error that could have played a role in the accuracy of our results. For example, when measuring the vertical and horizontal distances required to use the kinematics method, we used a meter stick that could not measure in hundredths. As a result, we were forced to estimate, leading to some calculations that may have been different than what they would have been otherwise. Another source of error could have been the way in which the photogate was positioned. Any tilt in its position could have easily thrown off the results of that method and given different, inaccurate velocities. To decrease this error, we would use a measuring tool that could measure hundredths of meters and we would have clamped the photogate down to the table it was on to make sure that it did not move.

Even though the ballistic pendulum is not a common scenario in real life, there are various examples in real life where objects only move in a ballistic fashion or have pendulum movement. A cannon is a recognizable example of ballistic motion, while a swing at a playground moves in a pendulum. However, inelastic collisions are very common, from car crashes to football players tackling each other.