Group2_4_ch6

toc Lauren Barinsky Lauren Kostman Maddie Margulies Nicole Tomasofsky

Lab: The Law of Conservation of Energy
2/3/12

__**Objective:**__ What is the relationship between changes in kinetic energy and changes in gravitational potential energy?

__**Hypothesis:**__ The changes in kinetic and gravitational potential energy should be equal at the initial and final positions. This is due to the Law of Conservation of Energy.

Station 1: media type="file" key="Movie on 2012-02-03 at 11.02.mov" width="300" height="300"

At the first station, we had to find the speed of the cart at the bottom of the incline. The initial and final height was measured. Also, the distance of the smallest part of the picket fence piece was measure because this was went through the photo gate. This was done, so we could record the time.

Station 2: media type="file" key="Movie on 2012-02-03 at 11.46.mov" width="300" height="300" First, the diameter of the ball was measured and the ball was launched at a short range. Then, the time in each photo gate was measured as the ball passed through them.

Station 3: media type="file" key="Movie on 2012-02-03 at 11.23.mov" width="300" height="300" At this station, the pendulum was let go at an initial height of 20 cm above the counter. The time was recorded of the cork when it was in the photo gate. The diameter of the cork was measured to find the velocity.

Station 4: media type="file" key="Movie on 2012-02-03 at 11.10 At this station, we had to find the maximum height of the ball at the top of the incline. First, we measured the height of the ball above the counter and then sent the ball down the ramp. Then, we measured the maximum height traveled from the counter top.

Station 5: media type="file" key="Movie on 2012-02-03 at 11.28.mov" width="300" height="300" At this station, we had to find the velocity of the ball as it left the launcer. First, we measured the diameter of the ball, found the time it took using the photo gate, and found its mass. Station 6: media type="file" key="Movie on 2012-02-03 at 11.17.mov" width="300" height="300" At the last station, we had to find the speed at the top of the loop of the roller coaster. The diameter and the mass was found of the ball and the height was measured from the countertop to the top of the ramp. The ball was let down the ramp and it went through the photo gate, which measured the time. This was used to find the belocity.


 * __Data:__**
 * __Personal:__**


 * __Class:__**













__**Sample Calculations:**__



__**Analysis:**__ Since the total initial energy should equal the total final energy, the following work equations were derived for each station. Since the purpose of this lab was to see if the total initial energy was actually equal to the total final energy, we used percent difference to determine how similar the two values were.

Calculating percent difference for Station 1:

Table of Energy Values:

__**Conclusion:**__ It was hypothesized that the total initial energy (KEi+GPEi) should equal the total final energy (KEf+GPEf). This was proven to be relatively correct; ideally, if there was no work done (which in some cases there was- friction served as workout), then this statement would hold complete accuracy. The results suggest that this does occur, for overall, the total energies were equal to one another at the beginning and end. There were several areas of error for each of the six stations within this lab activity. It’s possible that each time measurements were done, they weren’t exact, which would harm the results for the measurement were taken in centimeters and then converted to meters, yet if a smaller, more precise scale of measurement was used, the results could have been more accurate. Friction, which is a source of workout, plays a major role in the percent difference. This is because the hypothesis is assuming that there’s no work being done. But, in stations 1, 4, and 6, in which ramps and/or a track was used, there was friction that slowed the motion of the object down. Therefore, this causes the percent differences for these trials to much somewhat inflated in comparison to the others. For station 2, the horizontal launch, we assume the ball passes through the laser is the radius; but sometimes the launcher is inconsistent (it might fall through a chord of the ball- a smaller distance than the diameter). Therefore, the results for the velocity could be a little higher than they should have been. This lab activity is very applicable to everyday life. For example, the roller coaster, station 6, can be used to help engineers and those who design roller coasters at amusement parks. There must be a minimum speed at which the object must move in order to prevent it from falling off from the track. Another relatable station to everyday life was station 3, which had the pendulum. If a person were to be swinging on a swing, he or she would be able to calculate the velocity, as well as the amount of energy at different points in their motion.

Law of Conservation of Energy for a Mass on a Spring
Lauren B. Lauren K. Maddie M. February 10, 2012 __**Objectives:**__ __**Hypotheses:**__ The spring constant will be equal to each slope on the Force vs. Displacement graphs for each spring. The most flexible will have the smallest spring constant, while the tougher spring will have a larger spring force constant.
 * To directly determine the spring constant k of several springs by measuring the elongation of the spring for specific applied forces
 * To measure the elastic potential energy of the spring
 * To use a graph to find the work done in stretching the spring
 * To measure the gravitational potential and kinetic energy at 3 positions during the spring oscillation

__**Procedure:**__ __**Part A:**__

media type="file" key="Movie on 2012-02-10 at 11.20.mov" width="300" height="300" First, three springs were set up on a stand and an initial mass was put on the spring. This was set as zero and then the displacement was taken of each mass put on. Six different masses were hung on the spring and the displacement of the mass was measures. This was done on three different springs and for each spring there were five trials taken. With this data, the spring constant for each spring was found and a Force vs. Displacement graph was made.

__**Part B:**__ media type="file" key="Movie on 2012-02-10 at 11.38.mov" width="330" height="330"

For part B, a mass of .500 kg was placed on the red spring. It was pulled down and then it was let go. A motion detector was placed under the mass and spring to determine the displacement. This was found in data studio. The maximums, minimums, and in between were determined on three of the hills on data studio. Therefore, a position time graph was made and the data was found. __**Data:**__

__Part A:__



__Table of Spring Constant Values:__

__Part B:__

Point A is when the mass is at its lowest position, point B is at the equilibrium position, and point C is at the highest position.



__**Sample Calculations:**__

__**Analysis (discussion questions):**__
 * 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?**
 * 2) Yes, the data for the displacement of the spring versus the applied force does indicate that the data for the spring constant is indeed constant for this range of forces. Each line increases at a constant slope, which shows that the spring constant (k), was the saem for each of the ranges of forces.
 * 3) **How can you tell which spring is softer by merely looking at the graph?**
 * 4) Just by simply looking at the graph, you can tell which spring is softer based on the slope. The line that is the least steep (has the smallest slope), will be the softest. This is due to the fact that each time more mass was added to it, it's vertical position had the most displacement.
 * 5) **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.**
 * 6) Due to the Law of Conservation of Energy, the amount of energy remains constant through each stage of the spring's motion; this means that the initial amount of total mechanical energy is equal to the final total amount of mechanical energy. While at rest, there is only gravitational potential energy. When it's being pulled on, there's energy added on in the form of work, and while moving up and down, there's elastic potential and kinetic energy.

__**Conclusion:**__ For part A of the lab, the results were very close. This was because there was little room for error. For each spring, the percent difference from the actually spring constant was no more than 10 percent. The R squared value for each spring was .99 or higher and for the red spring, the results were perfect with a value of 1. The percent error for this lab was very low of 2%, which means the results were very good. A source of error that could have taken place in part A was measuring incorrectly. This was because the measuring stick could have been crooked, or the measurements might have been exact. This could have been improved by being careful and taking exact measurements by lining up the meter stick with extreme caution. In part B of the lab, the results were very good. The percent differences for points A, B, and C were 8.89%, 3.70%, and 5.19%. The source of error in part B of the lab could have been from the photo gate misinterpreting the spring. The spring could have been pulled down at an angle, which would result in it moving from side to side. This could have been avoided by making sure the spring was pulled straight down. This would affect the measurement of the velocity and in turn affecting the other components to calculate accurate results. Also, the sensor only records certain increments of time and if it missed the spring, the results may not have been perfect. The sensor could have missed the max point, so this is why multiple trials are conducted. They are done to prevent a large percent error. However, These results were good because the percent differences were fairly low which means the sum of the total mechanical energies of all three points were very close. The object of the lab was to see if those TME were equal and the final results were pretty close.

=Roller Coaster Project= Nicole Tomasofsky Lauren Barinsky Maddie Margulies Lauren Kostman

Rough Sketch (With Points): Side View

Top View:



(Side view):
 * Pictures:**

(Top view):

media type="file" key="Maddie M, Nicole T, Lauren K, Lauren B- Galileo's Shooting Star.mov" width="300" height="300"
 * Video:**

Galileo's Shooting Star is a fast paced, thrilling roller coaster taking each passenger on a galactic journey. The riders will feel like they are flying through they galaxy as they go through vertical loops, horizontal loops, and hills. The roller coaster is not just fun however, it is also educational. As the riders embark on their trip, they will hear a short history of how Galileo developed the first telescope which allows all of us to see stars like we do today. Galileo's Shooting Star is the number one attraction at this theme park!
 * Come for a Thrilling Ride!**


 * Description (physics content):**

For this project, we built a roller coaster, Galileo's Shooting Star, to study The Law of Energy Conservation and other aspects of physics that we have previously learned this year. The Law of Conservation of energy states that energy cannot be created nor destroyed; however, it can be changed into a different form. Ideally, the total energy in a roller coaster should remain constant assuming that no energy was lost as heat. However, this is not realistic for the roller coaster built for this project because it is made out of merely paper and it is not very stable, undesirably affecting the motion of the "cart" (marble) used. There is also friction between the marble and the paper that is not taken into account, and due to the friction force the marble loses energy as it continues down the roller coaster. A large amount of energy was lost especially during the loops and turns. As the marble moves down the roller coaster, it has an acceleration. This acceleration was needed to make it over many of the hills and loops. The initial dropping point of the roller coaster is relatively high and the first part of the track is steep, so the marble has enough velocity to make it through the vertical loop and over the 2nd hill. Going down this hill and through the horizontal loop allows the marble to again gain speed and make it over the 3rd hill.

The acceleration was solved for by using one of the many techniques from kinematics, the law of conservation of energy, or Newton's second law. Newton's Laws of motion were taken into consideration while building this coaster or else it would not have been successful. The first law states that an object will stay in motion they are already in unless an unbalanced force causes change. Our coaster follows this law because initially the ball is at rest, but is acted upon gravity to begin motion at the start of the coaster. It gains speed in some parts due to the gravity force, while in others its speed is decreased or its direction is changed due to the normal force of the paper pushing against the ball and the friction between the two. The second law states that force equals mass times acceleration. When the marble was roller down the coaster, opposing forces such as friction, gravity, and normal were acting upon the ball to determine the acceleration. The third law of motion states that actions have equal and opposite reactions, which is shown at the end of the coaster when the ball exerts a force on the wall, which would exert an equal and opposite force on the ball. When the ball is on the coaster, it is exerting a force on the paper. The 3rd law says that the paper will exert a force on the ball that is equal to and in the opposite direction as the force of the ball on the paper. The force that the ball is exerting on the paper is equal to a component of its weight (depending on its position on the coaster), and its apparent weight, the measure of the downward force, varies from position to position. For example, the apparent weight is larger at the bottom of the hill entering the vertical loop, while it is smaller at the top of the 2nd and 3rd hills. This is why you feel very light when riding over hills but very heavy on the bottom of loops. Although the your true weight is always going downwards (due to gravity always going downwards), your apparent weight actually goes upward when you are at the top of a vertical loop.

In order for the ball to reach the top of the coaster, power has to be taken into consideration. Power is the rate at which work is done. In order to calculate the power needed to reach the top of the roller coaster in 45 seconds, the work (force times displacement) is divided by the time (45 seconds). This work is equal to the Gravitational Potential Energy at the top of the coaster. At the end of the coaster, a spring is needed to be placed, to ensure a safe end to the ride. In order to calculate this Hooke's Law is needed. It states that the force of the spring is equal to the negative spring constant (k) times displacement of the compressed spring (x). Also, this used used to find the Elastic Potential energy of the spring. At the end of the coaster, the spring constant that was calculated was 23.37 N/m and the stopping distance we used to solve for this was .09; the distance of the star at the end of the coaster.

Circular motion is when an object, the marble, rotates along a circular path. At the start of the roller coaster, the marble moves down the initial hill with kinetic energy and accelerates into the vertical loop. Since the law of conservation states that energy only changes its form, the kinetic energy is changed into gravitational potential energy and then changes throughout the coaster. If the diameters were larger for the vertical and horizontal loops, the ball might not have made it around. It is important to have enough velocity to make it around the loops.

__**Data:**__

__**Key:**__
 * __Link to Excel Spreadsheet:__**


 * Sample Calculations:**
 * a. Theoretical Velocity/Theoretical Acceleration/ Total Energy/ Energy Lost**





Total Energy





(Power needed to get to the top in 45 seconds):
 * b. Theoretical amount of energy and power to get roller coaster rolling**
 * The total energy used was the average of all 10 trials.


 * c. Minimum speed at top of vertical loop**




 * d. Minimum height requirement of the first hill**
 * e. Energy dissipated at end of the ride to bring the roller coaster to a halt**


 * f. Theoretical spring system to stop the roller coaster car in case the brakes fail**



**Discussion of how designers address the uncertainties inherent in their calculations:** The uncertainties in our calculations are due to the lack of stability in our roller coaster. Theoretically, there should be no energy dissipated, however, when the roller coaster shook it lost energy. Also, this was due to friction. In a real life situation, a roller coaster designer could account for the uncertainties that might be present by making the ride extra safe or giving it "wiggle room" in the calculations in case an unpredictable factor comes into play that would make the ride unsafe or dysfunctional. This might be necessary, for example, because of wind. There will always be air resistance acting against the cart, but the force will fluctuate as natural wind occurs. The designer must address this issue by making sure the roller coaster is safe and functional with the presence of different degrees of air resistance coming from all directions.

A way to fix our issue of instability is to put more diagonal supports on the roller coaster and to rebuild it with a more stable material rather than paper. If paper and tape were to be used in the future, sturdier ones could be used, such as construction paper and masking or duct tape. We could also line the paper with tape once we have it in the desired position. These stronger materials would allow for more stability, and this would result with more precise and accurate results (since the ride wouldn't get as weak over time). We would also get more accurate results in our calculations if we took into account the friction between the ball and the paper and the air resistance acting against the ball.

Humans can endure between 4-6 g's while on a roller coaster. This is dependent upon the velocity at a given point, and the radius of the loop it is on/ amount of distance it's traveled. Our calculations confirm that Galileo's Shooting Star is in fact safe for humans to ride on, since they fall within this range of g's. In addition, the vertical loop is safe because it is going faster than the minimums speed, so the riders will not fall out of their seats.
 * Safety**:



Lab: Elastic and Inelastic Collisions
3/9/12

__**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?

__**Hypothesis:**__ Due to knowledge about the Law of Conservation of momentum, the initial momentums should equal the final momentum of a system. An elastic collision is one where kinetic energy is conserved, and an inelastic collision is one where kinetic energy is not conserved. Therefore, if there is a big difference between the total initial kinetic energy and the final kinetic energy then a collision would be elastic, but if there is a small difference between the total initial kinetic energy and the final kinetic energy, then a collision would be inelastic.

__**Materials and Method**__: For this lab, a metal track was set up with two motion detectors on each end and two carts on the track. Multiple collisions were then performed with different masses added to the carts each time. There was no constant, since the collision was started by hand, so many multiple trials were done of many different types of collisions. However, the law should be proven correct for all collisions. After the collision occurred, a velocity time graph was produced on data studio where the initial and final velocities were able to pinpointed. Analysis of the total initial momentum and the total final momentum can be performed using this data to find whether the collisions were inelastic of elastic.

__**Video:**__ media type="file" key="Movie on 2012-03-09 at 12.05.mov" width="300" height="300" (Cart A colliding into Cart B at rest)

__**Data**__: __**Excel Spread Sheet:**__ __**Sample Data Collection:**__

(This shows cart A colliding into cart B at rest)


 * __Sample Caluclations:__**

1. Total momentum 2. Total kinetic energy

3. Percent Difference
 * __Analysis:__**
 * 1) Is momentum conserved in this experiment? Explain, using actual data from the lab.
 * 2) Momentum is conserved in most of the trials (except for the Explosion ones, yielding a 100% difference) because the percent differences are very low, under 10%. The initial and final momentums are very similar. In the head on collision, the initial momentum was 0.140 and the final momentum was 0.145, showing a very small difference and leading us to conclude 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) When carts of unequal masses push away from one another, the cart with the smaller mass has a higher velocity because the force of the explosion pushes the lighter cart at a higher speed than the cart with more mass. Momentum is mass times velocity; this direct relationship means that these two magnitudes must balance each other out (when set equal to each other). As mass decreases, velocity increases.
 * 5) When carts of unequal masses push away from each other, which cart has more momentum?
 * 6) When carts of different masses push away from each other, the cart with the larger mass has more momentum; this is due to the equation p=m*v. The direct relationship means that as the mass increases, so does the momentum.
 * 7) Is the momentum dependent on which cart has its plunger cocked? Explain why or why.
 * 8) Momentum does **not** depend on which cart has its plunger cocked; this is because they both have the same force acting on them, just in different directions. The momentum is dependent on mass and velocity, not the plunger.

__**Conclusion:**__ The results obtained from this lab activity support the hypothesis that the total initial momentum is equal to the total final momentum. The trials that had large differences between their initial and final momentums were elastic collisions, meaning that the kinetic energy is conserved. Similarly, for the trials that had minute differences between their beginning and final momentums were inelastic collisions, where the kinetic energy was **not** conserved. For example, during a head on explosion, there was only a 2 percent difference, which suggests that it was an inelastic collision. The low percent differences suggest that the results were very precise. Nonetheless, there were several areas where errors could have occurred. For example, as masses were added to one of the carts, we didn't remeasure the total mass of the system, so it's possible that although we **assumed** that each mass weighed the same as the first one (.497 kg), however we didn't test this to make it certain. Additionally, it's possible that when the carts were given an initial push, it's possible that something (such as a hand) got in the way of the sensor, which would have impacted the results on Data Studio. If this lab were to be done in the future, there are several ways in which the procedure could be improved. One way is by using a cart that has a motor to collide into a regular (non-motor) dynamics cart so that the initial velocity can't be harmed by an outside force. This lab is extremely applicable to everyday life. For example, car accidents, collisions in sports, and other explosion/collision situations can be related to this activity, and the momentums can be determined.

Ballistic Pendulum Lab
__**Objective**__: What is the initial speed of a ball fired into a ballistic pendulum?

__**Hypothesis**__: The initial speed of a ball fired into a ballistic pendulum should be the same as the other trials that will be conducted in this experiment.

__**Videos/ Methods and materials**__: Pendulum Video: media type="file" key="Movie on 2012-03-16 at 11.06.mov" width="300" height="300"

First, we took the mass of the ball and the mass of the pendulum and ball together. Then, we set up the projectile launcher and attached the pendulum to it. The ball was shot into the pendulum while it pushed the angle indicator on the pendulum. This gave us the angle to be able to calculate the height of the pendulum and then we calculated the initial velocity.

Photogate Video: media type="file" key="Movie on 2012-03-16 at 11.11.mov" width="300" height="300" For this experiment, we set up the projectile launcher and shot the ball at short distance. The photo gate took the time of the ball in between the gate. Along with measuring the diameter of the ball, this data allowed us to calculate the initial velocity.

Projectile Video: media type="file" key="Movie on 2012-03-16 at 11.25.mov" width="300" height="300" For this experiment, we set up the projectile launcher and then shot the ball at short range to see where it would land. Then we put carbon paper on the ground and measured the distances from the launcher to where it landed. Then we look the average of these distances. We also took the height from the ground to where the ball is launched from. Using these measurements, we calculated the initial velocity by using projectile kinematics.

__**Data**__: Link to Excel Spreadsheet:

__**Sample Calculations:**__

Initial Velocity for Photogate:

Initial Velocity for Projectile:



Percent Difference __**Discussion Questions/ Analysis:** __ A perfect elastic collisions conserves kinetic energy. Inelastic collisions do not conserve kinetic energy. In an inelastic collision, kinetic energy is partially changed to some other form of energy. Maximum loss of kinetic energy would be an inelastic collision, like one when one object colliding with another object at rest and the center of mass does not have a change in velocity.
 * 1. In general, what kind of collision conserves kinetic energy? What kind doesn’t? What kind results in maximum loss of kinetic energy? **


 * 2. Consider the collision between the ball and pendulum. **

Inelastic
 * a.Is it elastic or inelastic? **

No, the kinetic energy is not conserved.
 * b.Is energy conserved? **

Yes, the Law of Conservation of momentum states that momentum is the same before and after collisions.
 * c. Is momentum conserved? **

No, the collision is inelastic therefore energy is not conserved. Like in the example above, an object hitting another object at rest without a change in velocity from the center of mass in inelastic. Kinetic energy is completely conserved.
 * 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? **

The momentum is conserved even though the kinetic energy is not, based on the Law of Conservation of Momentum.
 * b.How about 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. **

No, the collision is inellastic as confirmed through the 82.6% energy loss after the collision.
 * c. According to your calculations, would it be valid to assume that energy was conserved in that collision? **

<span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">Compared to .826, .788 is not that different. However there is discrepancy of about 4% error. The results are not bad though which is another way to measure that our data is pretty accurate.
 * <span style="font-family: Arial,Helvetica,sans-serif;">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. **

As the mass of the ball increases, the initial velocity increases. This makes the pendulum go to a larger angle and a higher height. But increasing the mass of the pendulum decreases the height and makes theta smaller.
 * <span style="font-family: Arial,Helvetica,sans-serif;">5. Go to [] Select “Ballistic Pendulum” from the column on the left. **
 * <span style="font-family: Arial,Helvetica,sans-serif;">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.) **

Our largest percent difference was about 10.2% which is relatively low but could be better. This is not a significant difference however. Our kinematics had the largest percent difference. This could come from the fact that kinematics required a lot of measuring, leaving a lot of places for error to occur. However, there should not be a large energy loss in the kinematics and photo gate method. A way to get better results would be to get a spring that allowed the shooter to be more precise. In addition, a better reading of theta, one that includes more decimal points will help. Finally, including air resistance in our calculations, something we initially disregarded would overall yield better results.
 * <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? **

__**Conclusion**__:

The three different methods we used all yielded similar results, the percent differences all being under 1.6 with the exception of one trial. The ballistic pendulum method gave us an initial velocity of 3.64 m/s with % differences under 1.449 with just one outlier of 2.634. The photogate method gave us 3.496 m/s with a maximum % difference of 0.647. The projectile method gave us 3.048 m/s with a maximum % difference of 1.575.

A source of error for the ballistic pendulum method came from the friction between the angle indicator and the surface it was on. As the pendulum moved up when the ball was launched into it, it pushed the angle indicator up with it so we could accurately measure the angle. If it weren’t for this friction apposing the motion of the pendulum, the pendulum might have been able to rise to a slightly higher angle, giving us a larger initial velocity. To minimize this error, we could use an electric motion-detecting device that indicates the highest position that the pendulum reached on the protractor.

A source of error for the photogate method came from the placement of the pohotgate. If the sensors were not lined up with the center of the ball, it probably recorded a smaller time that the ball was in the gate for, thus a larger velocity. Since we measured the diameter of the ball as the distance, the time used to calculate the velocity should be relative to the diameter. To minimize this source of error, we could attach the photogate to the launcher at the desired height instead of simply resting it on a calculator and trusting our eyes that the sensors are level with the center of the ball.

A source of error for the projectile method came from the position of the ball inside the launcher. When we load the launcher, we assume the ball is staying all the way back where we pushed it to, but it may have moved forward before we launched it, decreasing the amount of force on the ball. This could have given us a smaller initial velocity. To minimize this source of error, we could set the launcher at an angle to ensure that the ball cannot creep forward, and then calculate the velocity according to the set angle.

According to our results and analysis of the possible sources of error, we believe that the photogate method was the most accurate way of determining the initial velocity of the ball. There seems to be minimal sources of error for this method, partly because once everything is set up, our results are dependent on the computer, which I feel we can trust to make accurate measurements. The percent differences of each trial of this method were all very low, the highest being 0.647. Also, this method yielded an initial velocity of 3.496 m/s, which is in between the velocities yielded by the other methods, 3.048 and 3.64 m/s. This velocity is also much closer to the velocity from ballistic pendulum method (3.64 m/s) than the velocity from the projectile method (3.048 m/s), showing that the photogate method yields good results to be compared to those of the ballistic pendulum method.