Murtagh,+Miller,+Tosi

Lab 4: Ballistic Pendulum

Problem: What is the relationship between initial and final momenta of a ballistic pendulum?

Purpose: To discover the relationship between initial and final momenta of a ballistic pendulum.

Hypothesis: The final velocity of the ball and pendulum together will equal the initial velocity of the work-energy segment in this lab; however, the amount of kinetic energy when the ball is initially launched will not equal the final gravitational potential energy at the end due to the collision between the pendulum and ball being perfectly inelastic (they stick together). Because we know the law of conservation of momentum to be true in 2 dimensions (as proven in our glancing collisions lab), and we know that energy cannot be created or destroyed, as proven by scientists and by us in past labs, we believe that by comparing the velocities of these two segments that we can ascertain the validity of the law of conservation of momentum.

Materials:

Ballistic Pendulum Ball Textbooks Carbon Paper Laptop Excel

Procedure 1: (Finding initial velocity of ball)

1. Set launcher to medium velocity. 2. Place textbooks below launcher, and set launcher horizontally (angle = 0) 3. Launch ball to see how far it goes, then place carbon paper on place where it landed. 4. Repeat and measure different distances from different trials, then average them, and measure vertical distance. Use this information in order to find the velocity of ball using 2D kinematics.

Procedure 2: (Finding final velocity of momentum segment):

1. Use initial velocity calculated using projectiles, coupled with masses of pendulum (and weights) and ball, to calculate final velocity of perfectly inelastic collision. 2. Compare this value to velocity found in procedure 3.

Procedure 3: (Finding initial velocity of work-energy segment):

1. Set launcher to "medium speed" and place ball into launcher. 2. Launch ball just to see what the angle will be about. 3. Place arm at about that angle and launch again. 4. Repeat, placing the arm a little higher or lower than before depending on how much time the pendulum remained in contact with the arm (we want to minimize the contact time between these two). 5. Using trigonometry and work-energy to find out the velocity initially of the pendulum.

Final: Compare the velocities of work energy and momentum, and in theory, these two values should be the same.

Data:

Sample Calculations:





Discussion Questions:

**In general, on elastic collision conserves energy. An inelastic collision does not conserve energy. A perfectly inelastic collision results in maximum loss of kinetic energy.** >>> **It is inelastic.** >>> **Since the collision is inelastic, energy is not conserved.** **Momentum is theoretically conserved, but due to many sources of error, our data and calculations show otherwise.** >>> **The energy is not conserved because this collision is elastic and some kinetic energy is lost.** >>> **The momentum is conserved in this series of actions.**
 * 1) In general, what kind of collision conserves kinetic energy? What kind doesn’t? What kind results in maximum loss of kinetic energy?
 * 1) Consider the collision between the ball and pendulum.
 * 2)  Is it elastic or inelastic?
 * 1)  Is energy conserved?
 * 1) Is momentum conserved?
 * 1) Consider the swing and rise of the pendulum and embedded ball.
 * 2)  Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height?
 * 1)  How about momentum?

>>> **The average loss in kinetic energy before and immediately after the collision is (4.30-2.64=1.66).** >>> **The percentage loss in kinetic energy is 38.6% (1.66/4.3*100=38.6%).** >>> **According to my calculation, it would not be valid to assume that energy was conserved in that collision due to the high percentage of the initial kinetic energy that was lost after the collision.** >>> **The ratio of M/(m+M) is ((2.38/(2.38+.66))=??????**
 * 1) It would greatly simplify the calculations if kinetic energy were conserved in the collision between ball and pendulum.
 * 2)  Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum.
 * 1)  What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy.
 * 1)  According to your calculations, would it be valid to assume that energy was conserved in that collision?
 * 1)  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.


 * 1) Is there a significant difference between the two calculated values of velocity? What factors would increase the difference between these two results? How would you build a ballistic pendulum so that momentum method gave better results?

Conclusion: We hypothesized that since this lab produced a perfectly inelastic, KE would be lost after the equation, but momentum should remain somewhat the same. Though we did not get a percent difference of 0 for momentum, our percent different of 33.2% was less than the percent different of KE, which was 48.0%. This would make sense, because if we assume that much of the difference can be attributed to error, there would still be a loss of KE compared to a loss of momentum. Some of this error can most likely be attributed to imprecise measurements. When finding the initial velocity, we had to stack the cannon on textbooks, so that we would have an easier measurement. However, because of the shape of the books, we had to break both our x and y measurements into choppy pieces and add them, so there was room for human error there. Also, the small arm the pendulum pushed to indicate the angle would counteract the movement of the pendulum, absorbing some of the momentum - so the momentum was not lost, as we predicted, but misplaced. Also, we were not given many significant figures of measure for the angle, so those readings were somewhat imprecise. to improve this lab, we could have used some higher tech device to find a more accurate initial velocity (like a photogate). We could have also used measuring devices that gave us more precise readings.

Lab 3: 2-D Collisions with Hover Pucks Date Assigned: March 30th


 * Problem: **
 * Is the law of conservation of momentum true? Does the amount of momentum remain the same before and after a collision?


 * Purpose:**
 * To verify the law of conservation of momentum by comparing the amount of momentum before a collision and after, using 2-D motion (velocity and angle)


 * Hypothesis:**
 * Because of the law of conservation of momentum, which states that in an elastic collision, momentum will not be gained or lost, the net momentum, when you add the momentums in the x and y directions will equal the momentum before the collision (m1v1f+m2v2f). In our last experiment, momentum was essentially preserved after the collision, so we have no reason to suspect that if the final motion is in 2 dimensions, the findings will be any different.


 * Materials:**
 * 2 Hover Pucks
 * Tape measure
 * 3 stop watches
 * Laptop
 * Excel


 * Procedure:**
 * Mark the exact starting points of Hover Puck A and Hover Puck B
 * B should start at rest
 * Push A from starting point towards B - timing the trip from start to exact collision point
 * Time trip of A from exact collision point to wherever stopped - leave A at exact stop point
 * Time trip of B from exact collision point to wherever stopped - leave B at exact stop point
 * Measure distance from A starting point to B starting point (AKA collision point), and divide by time to find velocity of A before collision
 * Measure distance from B starting point (AKA collision point) to wherever A was stopped, and divide by time to find velocity of A after collision
 * Measure distance from B starting point (AKA collision point) to wherever B was stopped, and divide by time to find velocity of B after collision
 * Calculate initial x momentum (velocity x mass of A)
 * Calculate final x momentum (x momentum of A + x momentum of B)


 * Data:**

(Sorry but the picture was too long to be fit in one file. The table can be read straight across and the first picture continues to the second picture and the second picture continues to the third picture.




 * Sample Calculations:**







Scaled Drawing of Collision:





**Discussion Questions:** 1. In general, what kind of collision conserves kinetic energy? What kind doesn’t? What kind results in maximum loss of kinetic energy? **Elastic collisions conserve kinetic energy, while inelastic collision does not. A perfectly inelastic collision would produce maximum loss of kinetic energy.**

2. Consider the collision between the two hoverpucks. a. Is it elastic or inelastic? **Inelastic**

b. Is energy conserved? **Energy is transferred into another form (lost as heat due to friction)**

c. Is momentum conserved? **Momentum is theoretically conserved, based on our low percent differences with a multitude of errors possible**

3. It would greatly simplify the calculations if kinetic energy were conserved in the collision between two hover pucks. a. Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision. **The loss of kinetic energy in the first trial is .804J.** b. What is the percentage loss in kinetic energy? Find by dividing the loss by the original kinetic energy. **The percentage loss in kinetic energy 65% (((1.236-.432)/1.236))*100=65%).** c. According to your calculations, would it be valid to assume that energy was conserved in that collision? **According to my calculations, it would not be valid to assume that energy was conserved in that collision because 65% of the initial kinetic energy is lost.** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10px; margin-bottom: 0px; margin-left: 1.25in; margin-right: 0px; margin-top: 0px; padding: 0px; text-indent: -0.25in;">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. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10px; margin-bottom: 0px; margin-left: 1.25in; margin-right: 0px; margin-top: 0px; padding: 0px; text-indent: -0.25in;">**The ratio of M/(m+M) is (.705)/(.705+.301)=.70. This is relatively similar to the percentage lost in kinetic energy, .65. These qualities seem to agree with each other.** <span style="font-family: Arial,Helvetica,sans-serif; font-size: 10px; line-height: 15px; margin: 0px; padding: 0px;">4. What assumptions did we make that may affect our results? How would you change this lab to address these issues? In this lab, we assumed that the hoverpucks moved at constant speed, meaning we assumed no friction. Also, we assumed our short distances and small times were accurate. In the future, we could use a more open area and a surface with less natural friction so that would could increase the distances and time to get a more accurate measurement of speed, and therefore angles. Also, we could measure the distance from the center of one hoverpuck to the center of the other, similar to the distance between planets as we studied in gravitational motion.


 * Conclusion:**

In this lab, we investigated if momentum would be conserved in 2D collisions. We hypothesized that because momentum was conserved in 1 dimension, momentum would be conserved in two dimensions as well. Specifically, we took a hoverpuck and hit it at an angle so that it would “glance” the other hoverpuck at rest, bumping each of them at a certain angle and speed. We calculated the momenta of the x and y axis before the collision (pi) and the momenta of x and y components after the collisions (pf) and compared the two values using percent difference. Our data indicates approximately a 14.69% difference at best and 41.23% difference at worst. With these relatively low numbers (given the room for error in this lab) and our knowledge of the law of conservation of momentum, we can conclude that this law is indeed true.

We also compared the KEi and KEf to discover if the collision was elastic or inelastic. Our enormous percent differences of 58% and 96% indicate a difference in kinetic energy. We do realize that these are inaccurate numbers; however, though extreme, this numbers still indicate that the collision between these two hoverpucks were inelastic, meaning kinetic energy was lost. We are unsure of how much was lost due to heat because of the inaccuracy of our numbers; however, we do know that some proportion of energy was lost based on our logic that elastic collisions are almost impossible to achieve in reality. We are choosing to look at this part of the lab qualitatively instead of quantitatively.

The error in this lab stems from the timing and the distances, which were used in order to find the velocities and angles in this lab. In terms of the velocity, we assumed constant speed because we assumed a frictionless surface; however, nothing is really frictionless. Also, in order to find the time, we used stopwatches, and because human reaction time is horrible in terms of short times, a small difference (a tenth of a second) could have resulted in a large error. For example, in our first trial, the time for the first segment was 1.02 seconds. If that time were really 1.12 seconds, we would have obtained a greater velocity than the actual velocity, throwing our results off slightly.

The optimal case for measuring the distance between hoverpucks would be to measure from the center of one to the center of the other; however, due to the hoverpucks’ natural inclination to spin and move slightly as a result of that spinning, we could not pinpoint a center, and therefore measured as close as we could to that center. Because the distances were measured this way, the angles we calculated using trigonometry were affected as well. Scientists use momentum concepts in two dimensions to reconstruct complex car crashes in order to determine who was at fault. Combing the law of conservation of energy, the law of conservation of momentum, and logic, these physicists make great use out of the concepts we have studied in order to prove beyond a reasonable doubt guilt or innocence of a person involved in a car crash.

Lab 2: Conservation of Momentum Date Assigned: March 25th


 * Problem: **
 * Is the law of conservation of momentum true? Does the amount of momentum remain the same before and after a collision?


 * Purpose:**
 * To verify the law of conservation of momentum by comparing the amount of momentum before a collision and after.


 * Hypothesis:**
 * Because of the law of conservation of momentum, the net momentum (m1v1+m2v2) will equal the momentum after the collision (m1v1f+m2v2f). We hypothesize this because we will know the masses and velocities before the collision, and the masses and velocities after the collision using data studio, and we will compare these values in order to prove the law of conservation of momentum. Also, we know the law of conservation of energy to be true, and we know that the derivative of kinetic energy is momentum. Based on this relationship, we infer that the law of conservation of momentum is true.


 * Materials:**
 * Frictionless track
 * 2 carts
 * 2 motion sensors, connected to datastudio
 * Laptop
 * Various masses
 * Excel


 * Procedure:**
 * For this lab, we will be utilizing 2 motion sensors, hooked up to datastudio on our laptops. Each sensor will be on either side of a presumably frictionless track, facing towards the track. We will have 2 cars on the track, and will collide them in each of the following ways:
 * Explosion (both at rest, then moving at the end)
 * One at rest, and the other in motion – sticking together
 * One at rest, and the other in motion – bouncing apart
 * Both in motion – sticking together
 * Both in motion – bouncing apart
 * We will also alter the masses of each of the cars to change the ratio
 * We will record the masses of each of the carts for each run, and find on datastudio, their velocities directly before and directly after the collision
 * We will compare the net momentum before the collision to after the collision, and will hopefully find the same result


 * Data:**






 * Sample Calculations:**

Momentum:



KE:



Percent Difference:




 * Discussion Questions:**

1. Is momentum conserved in this experiment? Explain, using actual data from the lab.

Based on our high percent differences, we cannot conclude that in this experiment, momentum was conserved. For example, for one trial, we had approximately 20% difference between the initial and final momenta. We attribute this to multiple factors discussed in the conclusion.

2. When carts of unequal masses push away from each other, which cart has a higher velocity? Explain why this is.

The cart with the lower mass has a higher velocity. If momentum is held constant (as the law of conservation of momentum states), then if the mass decreases, the velocity increases, based on the equation p = mv.

3. When carts of unequal masses push away from each other, which cart has more momentum?

The carts should have equal momentum. Due to the law of conservation of momentum, initial momentum will equal final momentum. Because initial = 0, the momenta of the two carts should equal each other. Therefore, the cart that has more mass will move slower than the other, thereby "balancing" the difference in mass.

4.Is the momentum dependent on which cart has its plunger cocked? Explain why or why.

It does not; the plunger only increases the contact time of the two carts, thereby decreasing the force between them, but because momentum cannot be created or destroyed, the momentum remains unchanged.


 * Conclusion:**

In this lab, we did not effectively prove the law of conservation of momentum to be true. Though we know this law to be true, due to our relatively high percent differences between initial momentum and final momentum, (for example, in the 1 car moving bounce trial, we obtained a 22% difference between the initial and final momenta), we could not conclude from our data that this law is true for all but explosions.

In terms of the explosion, we could not calculate percent difference due to the fact that initial momentum was 0, meaning the formula for percent difference would yield the same result each time: 200% difference. In order to combat this, we compared the momentum of one cart at the end to the momentum of the other cart at the end, as well as the kinetic energy of one cart to the kinetic energy of the other. By looking at the data, we saw that the values we compared were very close to each other, and therefore, we were able to conclude the law of conservation of energy was true for explosions.

However, we also calculated the change in KE, where we saw a very minute difference between the KEi and KEf, effectively proving the Law of Conservation of Energy, and since momentum is the derivative of kinetic energy (how fast the kinetic energy is changing = momentum), then we can understand that the law of conservation of momentum is true, despite our inability to effectively prove it.

Our high percent differences could have resulted from a multitude of factors. We assumed the track to be frictionless; however, frictionless is a concept impossible to achieve. Due to the fact that the carts were moving relatively slowly, the friction would have affected them much more than if they were moving at higher speeds. The presence of friction would have resulted in the kinetic energy being lost due to heat or work due to friction. Also, the masses we used were not exactly the same mass, which would have affected our initial and final momenta calculations. Once again, due to the fact that the velocities were relatively small, a slight difference in the masses would have created a substantial difference between initial and final momenta. In the future, we would try to simulate a situation where we could push the cart at a higher velocity so that friction's effects would be lessened. Also, we would make the carts as close to the same mass as possible, instead of assuming that they were.

Lab 1: Crush Energy Date Assigned: March 21st Date Due: March 25th

Problem:

How much energy causes an aluminum can to crush a certain distance? How can we apply this concept to automobiles?

Purpose:

To estimate the Crush Energy from Damage Measurements on an aluminum soda can, which will serve as a model for an automobile, and then compare these measurements to other concepts we have studied (work-energy, kinematics).

Hypothesis:

Energy due to Crush found through estimation should equal kinetic energy found using work-energy concepts and kinematics.

Materials:
 * Two Coca-Cola Cans
 * Ruler
 * Aluminum Ball
 * Notecard
 * Excel

Procedure:
 * 1) Measure the distance from the ball height to the top of the can.
 * 2) Drop the ball onto the can.
 * 3) Measure the width of the dent.
 * 4) Cut a notecard to fit into the dent.
 * 5) Divide the cut notecard into five sections.
 * 6) Using constants, find crush energy for one section.
 * 7) Sum the crush energies to find the total crush energy.

media type="file" key="jt cm am clip crush.mov" width="300" height="300"

Pre-Lab Calculation:



Data:









Sample Calculations:





Conclusion: Though it proven that the amount crush energy will equal the net amount of kinetic energy before the crash, we were unable to verify this with this lab. The error could have stemmed from many places. First, the conversion factor that we used (x 30) could be a problem. We do not know about the different between aluminum cans and functioning cars to conclude that 30 is an appropriate and mathematically accurate conversion factor. We do not know if 30 compensates for the difference in size and stiffness. Also, we are unsure if we used the correct wheel base measure, and the constants "A" and "B" that corresponded with that. We don't know enough about where this specific measure came from to conclude that this appropriate to use for this experiment. Finally, the way we were able to measure the depth of the crushed section was by roughly cutting a piece of paper the shape of the indent. This method is not very precise, and allows much room for discrepancy and error. This method could have skewed our results.

Besides the multiple "kinks" in this lab that would need to be straightened out to make this lab produce the proper result, there are a few ways to improve it beyond that. Using a different method than paper cutting to recreate the dent more precisely would improve the results. Also, cutting the paper into more intervals would make the measurements more precise. Lastly, if we needed to solve for the kinetic energy before the crash, it would be suitable to use work energy or EPE instead of crush energy, because we know that would produce an acceptable result in this case.