19-April-2017: Impulse-Momentum

Lab 14: Impulse-Momentum

19-April-2017
Idea: Verify the impulse-momentum theorem by observing collisions in which we observe a cart to simulate an inelastic or elastic collision.

Summary:

EXPT 1:

1. We fasten a cart-track system as shown below. There is a rubber stopper on the cart connected to our force sensor, and once given a pull via a wire, meant to hit and bounce off of an obstacle at the end of the track. In this lab, another cart with a plunger.
   
2. Then, we have a motion sensor on the opposite end of the track, and in Logger Pro make sure that the position towards the motion sensor is the positive direction.
3. After calibrating and zeroing our force sensor, we give the cart a gentle pull towards the stopper multiple times until we get a good set of graphs. These graphs give us Force vs Time and Velocity vs Time, which we need in order to find Momentum, and consequently demonstrate the impulse-momentum theorem.

EXPT 2:

1. With the same setup, we add 500 grams to the cart and repeat the experiment, in order to compare whether the impulse and change in momentum were still equal to each other with a larger cart.
2. Thus we obtain a similar graph to EXPT 1 and compare their results.

EXPT 3:

1. Leaving the extra mass on the cart, we replace the cart at the end of the track with a vertical piece of wood with a lump of clay attached onto it. This experiment is meant to simulate an inelastic collision, wherein the cart stops immediately after collision.
2. I again obtain graphs of Force and Velocity, and calculate the impulse-momentum theorem.

Analysis:

EXPT 1:

Question 1: The net force exerted on the cart just before it starts to collide must be the force exerted by the spring on the extended cart's plunger, which increases as it compresses.

Question 2: The magnitude of the force is at its maximum when the rubber stopper's maximum compression of the plunger 

Question 3: After the collision, it is again the opposing force exerted by the plunger on the rubber stopper.

Question 4: The collision takes approximately a tenth of a second.

Question 5: *Calculations explained below*

Question 6: *Conclusion of EXPT1 explained below*

1) Our Force vs Time graph allows us to find the impulse. We can also calculate the momentum using the data recorded by the motion sensor, using p=mv. This allows me to demonstrate the impulse-momentum theorem by finding the change in momentum, and showing that it is in fact equal to the momentum applied to it.

2) Upon first inspection, on impact, the velocity suddenly increases to zero until it begins travelling in a constant positive direction towards the motion sensor.

3) Our graph gives an Impulse of .5290 N·S. Following my calculation of the change in momentum, I get a value of .560 N·S, relatively accurate to that of the graph, with a percent error of approximately 5%, very accurate. Therefore, I could reasonably determine that this adequately demonstrates the impulse-momentum theorem.

EXPT 2:

Question 7: *Conclusion of EXPT2 explained below*

1) Again on impact, the velocity suddenly increases to zero until it begins travelling in a constant positive direction, towards the motion sensor.

2) In order to demonstrate that regardless of the mass, the change in momentum is equal to the impulse, we again find the impulse and change in momentum of our new graph. Doing so produces an impulse of .841 N·S, and a change in momentum of .892 N·S. This again is accurate to a percent error of approximately 5%, so this indeed demonstrates that regardless of the mass, the impulse-momentum theorem holds.

EXPT 3:

Question 8: *Conclusion of EXPT3 explained below*

1) This time, the velocity does not continue to increase after 0. Because the cart sticks to the clay, the whole system stops. However, I reason that because momentum is still conserved, the impulse still equals the change in momentum.

2) To demonstrate this, I find the impulse and calculate the change in momentum for the last time, and I get values of 0.4314 N·S and 0.455 N·S, respectively. These values are very close to one another once again within 5%, and I can thus reason that even in the inelastic collision the calculated momentum equals the measured impulse applied to it.

Calculate Data

Results for change in momentum for experiments 1, 2, and 3.

Graphs:

EXPT 1
Impulse: .5290 N·S∆Momentum: .560 N·S

EXPT 2Impulse: .841 N·S∆Momentum: .892 N·S

EXPT 3Impulse: 0.4314 N·S∆Momentum: 0.455 N·S

Conclusion:

As demonstrated in experiments 1, 2, and 3, inelastic or elastic, we demonstrated that the impulse is in fact equal to the change in momentum, as for all three cases our calculations showed just how close the two values were. Therefore, we can reasonably conclude that this indeed demonstrates the impulse-momentum theorem. Between the three experiments, my calculations for percent error yielded 5.5%, 5.7%, and 5.2% respectively, showing that indeed this is a very reasonable conclusion.

• As for sources of error: the scale was inaccurate to .1 grams, and although it isn't very significant,  it is important because it does affect our change of momentum calculations to a certain extent.
• Our force sensor is approximately inaccurate to about .001 Newtons, as even in a zeroed state it would randomly read values like .0021 N. This would clearly lead to an inaccurate Force x Time graph, and ultimately an inaccurate value of Impulse. In addition, we had to make sure that the force never exceeded 10N, because apparently the sensors would read inaccurate data.
• In addition, similar inaccuracies to the motion sensor would create an erred velocity x time graph, inevitably leading to a value of a change in momentum with an uncertainty.
• Loss of energy. Clearly, these were not perfect elastic or inelastic collisions, as energy is inevitably lost due to friction, and for all three experiments this means that there is an error in both Impulse and Momentum, Friction would affect the cart's change in velocity.