15-April-2015: Impulse-Momentum Activity

The purpose of this lab was to prove the Impulse-Momentum Theorem: The change in momentum is equal to impulse.

Part 1: Elastic Collision:
Setup:

Fig. 1

To set up this lab, we needed a few materials:

• Two carts, one with an extended spring.
• A track that would direct one of the carts.
• LoggerPro, force sensor, motion sensor.
• Rod and some clamps.

First, we secured the cart with the extended spring to the lab bench using the rod and some clamps. Once we had done this, we setup up the track into such a way that a cart moving on the track would collide with the stationary cart attached to the lab bench. Once we had this, we attached the motion sensor to the cart which would move on the track, facing the direction of impact. Finally, we hooked up a motion sensor at the opposite end of the track (see Fig. 1 for set up).

Conducting the Experiment:

Once we had our setup complete, it was time to begin the experiment. First, we zeroed our force sensor in both the horizontal and vertical direction. Once this was done, and LoggerPro was ready with both sensors, we began collecting data and gave the cart a gentle push towards the stationary cart. The cart collided and repelled back.

Analyzing Data:

To test the impulse-momentum theorem, we first had to calculate the change in momentum. To do this we first found used logger pro to find the initial velocity of the cart and the final velocity of the cart. By taking the mean value of the velocity of the cart before and after the collision, we would be able to find both initial and final momentum of the cart (see Fig. 2). The mass of the cart we measured to be .678 kg. Thus the change in momentum = mv(f)-mv(o). = .678(-0.3965-0.4796)= -0.594 kg*m/s.

Fig. 2
After we had completed the lab, we found that one of our runs had been misplaced. During lab, we obtained the real values for initial and final velocity, however the graph from which we obtained it from had been deleted or altered. Thus, Fig. 2 contains a sample graph of what it the velocity graph for the first trial should have looked like. THIS IS NOT THE ACTUAL GRAPH.

For Impulse, we took the area under the Force vs Time graph (since we knew that Impulse= F*t, or the integral of F*dt.) and found that to be -0.6137 N*s. (see Fig. 3) With a 3.2 % difference, we could confirm the impulse-momentum theorem (see Fig. 4 for all calculations).

Fig. 3
Since we did not reverse the direction of the force sensor, we obtained a negative force, which in turn gave us a negative impulse.

Fig. 4
We compared our calculated impulse to that of LoggerPro's and found that the two values were reasonably close.

Part 2: Elastic Collision with More Mass:

Fig. 5

It was now time to conduct the experiment with the same setup, only now we added more mass to the moving cart. The goal of this trial was to determine whether or not the mass would affect the overall outcome of the impulse-momentum theorem.

With the exact same set up, we simply added 0.2 kg to the cart (see Fig. 5). Once again, we gave the cart a slight push, and collected data using LoggerPro. 

We found the initial and final velocity of the cart, which in turn lead us to the initial and final momentum of the cart. We took mean of the velocity graph before and after the collision which gave us an estimate of both the initial and final velocities. Our new cart mass was .878 kg. However, in this trial we added a magnetic mini white board to the back of our cart (this was to allow the motion sensor to track the cart more easily). The mass of the mini white board was .072 kg, which lead to a new total mass of 0.950 kg. We calculated our change in momentum to be -0.883 kg*m/s.

We again used LoggerPro to integrate the area under the Force vs Time graph and found the impulse to be -0.8537 N*s. There was 3.2% difference between the values. (see Fig. 6 for graphs, and Fig. 7 for calculations). This again confirmed the Impulse-Momentum Theorem.

Fig. 6
Since the velocity was relatively constant before and after the collision we were able to find the mean of the two horizontal pieces to find a reasonable value for momentum.

Fig. 7
Again, we were able to confirm the impulse-momentum theorem due to such a minimal difference between our change in momentum and impulse.

 Part 3: Inelastic Collision:
For the third portion of the lab, we wanted to examine the impulse-momentum theorem under inelastic conditions. To do this, we change our setup slightly. We replaced the stationary spring cart with a secured wooden block. Attached to this wooden block was a piece of clay. We then replaced then added a rubber stopper to the end of our force sensor, to which we attached a nail.The nail would then attach to the clay upon impact, not allowing the cart to bounce back. This would make our inelastic collision. We left the motion sensor and track as was (see Fig. 8).

Fig. 8

Once all of our equipment was set up, we once again gave the cart a slight push and began collecting data. With the cart being stuck to the clay upon collision, we knew the final velocity would be zero. Thus, we were only left to find the initial momentum. Since our graph gave us a slight downward slope, we tried to use the velocity most near the moment of impact, since this would be considered the initial velocity (see Fig. 9).

Fig. 9
Since the velocity is not reasonably constant, we measured the velocity 
nearest the point of collision.

Once we had this, we calculated our momentum. For this trial, we left the added mass on the cart and added the mass of the nail and rubber stopper which totaled to be 0.967 kg. We found our initial momentum to be 0.3733 kg*m/s, which meant our change in momentum was -0.3733 kg*m/s. We once again used LoggerPro to calculate the impulse exerted on the cart (see Fig. 10).

Fig. 10
We again used LoggerPro to calculate impulse using our Force vs Time graph.

We found this to be -0.305 N*s. (Note that we did not include the integral beyond the initial impact. The other humps in the graph were due to the vibration of the cart once it initially hit the nail. We compared the impulse we calculated to the impulse LoggerPro calculated and found that the values had a 20.1% difference (see Fig. 11 for all calculations).

Fig. 11
For our inelastic collision, we found that the difference between our change in momentum and impulse was significant. This indicates that there was some possible source of error. This could be explained perhaps due to the fact that velocity was not completely constant and there was some significant outside force(friction) acting on the cart.

Error, Comparison, and Conclusion:

When comparing the Force vs Time graph of the inelastic and elastic collisions, we found the they were similar in height, but different in thickness. Since we kept the extra mass on the cart for the inelastic collision, we compared the Force vs Time graph in Fig. 10 to that in Fig. 4. The thickness of the elastic collision force curve was much larger than that of the inelastic. 

There were quite a few areas of uncertainty in our lab. For one, we could not get any of the trials to a perfect constant initial velocity. This was especially apparent in our inelastic trial, in which the slope of the velocity graph is extremely apparent. However, perhaps the most prominent source of error was our inability to create a perfectly inelastic or elastic collision. This prohibited us from obtaining perfect results.

Once completed with the lab, we had verified the impulse-momentum theorem for both elastic and inelastic collisions. The elastic collisions had a small percentage difference between the different impulses, while our inelastic collisions had a rather large one. However, we believe that the larger gap may have been due to the fact that we did not give the cart a sufficient enough push and the added fact that the cart vibrated back and forth after colliding with the clay.