24-April-2017: Collisions in two dimensions

In the previous study, it is easy to know that momentum are always conservative in one dimension since we can prove it with the function

With the Newton's third law, they always have the same magnitude but opposite force, and the time will be equal. Thus, their momentum is conservative. However, does it still work in two dimensions or more dimensions. Today, we did a lab about the collisions in two dimension.

Purpose 

Look at a two-dimensional collision and determine if momentum and energy are conserved.

Plan

Find a smooth plane, and level it (as following) to decrease the influence of friction force and gravity.

Fix a camera with slow mode on the top.
Then, we collected two videos when a steel ball hit a steel ball, and when a steel ball hit marble ball.
Next, use Logger Pro to analyze the data of balls, and get the velocity.
Plug them into the functions of momentum and energy to see whether they are conservative.

Set up

Gently set up the stationary ball on the leveled glass table.
Then, level the table.
Aim the rolling ball so that it hits the side of the stationary ball, and collect the video by iPhone.

Experiment 1

Before collision

 After collision

Experiment 2

Before collision

After collision

Plug the videos into Logger Pro. Set the frame rate to 120 and advance the movie 4 frames after adding a new point.
Collect the points and set the length of the table to get the data of velocity.

Experiment 1

Moving steel ball's x and y-direction displacement  

Motionless steel ball's x and y-direction displacement

Experiment 2

Steel ball's x and y-direction displacement

Marble ball's x and y-direction displacement

Weight the marble ball and the steel balls.

Analyze

For momentum, we can divide them into x-direction and y-direction.
And use the function

For energy, we can calculate the velocity of each ball the directly use the function

For the first one

For the second one

By doing the subtraction, we can know that the differences and the changing percentage are

Experiment 1

Experiment 2

Thus, we can roughly make an assumption that momentum is still conservative because all the most momentum change is within 7%; while the energy is not since both experiment's energy changed over 30%.

Conclusion

For the collision in two dimensions, it still obeys the law of momentum conservative, since most change is with 7%. Except we will make some mistakes on collecting data (it is hard to avoid), the main reason for that is when the collision happens, there is a part of the energy which turns into heat. Moreover, because the balls are not smooth, they might be spinning after collision, and the span would make a larger friction with the plate, which will produce more heat.
In fact, even in three dimensions, the collision still has momentum conservative. Because we are detecting the objects with low speed, which implies that they follow the Newton's Laws. Then, for each direction,

then, the momentum will not be created or disappear. 

However, there is still some energy consumed since both experiment's energy is changed at least 30%. It is easy to know that there is some friction force between balls and plate, which will produce some heat (cost energy). Moreover, when they collided, they would also produce some heat because they are not elastic, and they will comsume some energy in deformation. 

In this lab, there are several factors that maybe affect the data. 

The first one is whether the table is well leveled. If it is not leveled, the gravity will participate as an external force and affect the impulse and momentum. Moreover, it always has some friction force that will influence the momentum as an external force. 

The second one will be the points collection. It is hard to hit the same place of the ball over and over again. We might hit the point next to it, which might affect the velocity. 

The third one will be the camera. There is a possibility that the camera is not leveled. If it is not, then it will slight influence the result because the distance measure will be different from the parallel one.