19-April-2017: Impulse-Momentum

It is very easy to discover that there is relationship inside force, time, mass, and velocity. When we push an object with a larger force or a longer time, the velocity will increase larger; in the meanwhile, if the object has a larger mass, then it will gain a smaller velocity. Thus, velocity has a positive relationship with force and time, and it has a negative relationship with mass. So we can make a conclusion that Ft/M = v or Ft = Mv. However, how could we know whether it will really apply to the object? In today's lab, we study the relationship inside the impulse and momentum.

Purpose

Observe and verify the impulse-momentum theorem.

Plan

Observe two different kinds of collisions. One of them is using a stopper, which gives a longer collision and then use a larger mass to do the experiment again. The other one is using a nail to poke into clay to give us a shorter collision (as the diagram showing in the following). 

Then use the integral to calculate the impulse, and use the velocity times mass to calculate the momentum. 

Then, compare impulse and momentum. If the function applies to the objects, then the impulse and momentum should be equal. 

Last, use the functions in Newton's system to prove the relationship. 

Set up

Fasten the force probe securely to the cart so that the stopper extends beyond the front of the cart.

Set up the motion detector as shown. Level the ramp. 

Determine what quantities you will need to measure to do the experiment. 

Open a new file, and set a graph's y-axis to be momentum. 

Set the positive direction is toward left. Next, calibrate the force sensor. (Because we do not have a pulley, in this case, we need to calibrate it horizontally. Then zero it in the vertical direction.)

Next, click the collect and push the cart to collect data. 

Then, we added a 200-gram mass on the cart and did the experiment above again. 

After that, we left the 200-gram mass and switched the stopper into a nail to get a shorter collision. 

Open a new file, and set the y-axis into impulse and momentum. 

Calibrate the force probe again. Zero it after we put it down. 

Click the data and push the cart, we can get a data of inelastic collision. 

Analyze

In the first experiment, the integral of force-time is 0.4672 N*s. The beginning momentum is 0.239 kg*m/s, and the final momentum is -0.193 kg*m/s, which means the changed of momentum is 0.432 kg*m/s. The difference is 0.035, which is 7% of the data. 

In the second experiment, the integral of force-time is 0.8557 N*s. The beginning momentum is 0.378 kg*m/s, and the final momentum is -0.390 kg*m/s, which means the changed of momentum is 0.768 kg*m/s. The difference is 0.0877, which is 10% of the data. 

These two examples are quite not what we expected. The errors are too large for the conclusion. I think the main reason will be we did not level the track correctly. Moreover, when the cart bounces backward as free body diagram shown in the following, the gravity and the friction were also doing work on it. 

In the second experiment, the integral of force-time is 0.2418 N*s. The beginning momentum is 0.237 kg*m/s, which means the changed of momentum is 0.237 kg*m/s. The difference is 0.0048, which is 2% off. 

Because this time, it did not bounce backward so the error will be lower.

Overall, it is not hard to find out that all of three experiments have the largest force in the medium of collision. In my opinion, no matter the stopper or the clay will have a short time of deformation. In the stopper case, it will give the same result as spring. In the clay case, in the beginning, the contact area between nail and clay is increasing, so that the relative force of deformation will increase. In the later part, because the relative speed is lower, the speed of deformation became slower, and it gave smaller force. 

Proof

Because in this lab, we were studying objects in low speed, which means all the objects are satisfied with Newton's Laws. Then, the relationship can be 

Thus, the change of impulse is equal to the change of momentum. 

Conclusion

As the result showed in the lab, we can say that the change of impulse is almost equal to the change of momentum. However, because there are some other factors (the friction is one of the largest reasons), the result came out that the change of momentum is lower than the change of impulse. Even though, we balanced the friction force, when the cart bounces up, the part of gravity and friction force still will affect the data. 

As a result, my lab was kind of failed because the second data is over 10% error. Nonetheless, the relationship between impulse and momentum still exists. First reason is our other data came out that there was less error. Second reason is this relationship can be proved by mathematic under Newton's system. 

Therefore,