24-April-2017: LAB 15: PHYS 4A LAB: Collisions in two dimensions

Eric Chong

LAB 15: PHYS 4A LAB: Collisions in two dimensions

Lab Partners: Lynel Ornedo, Nina Song, and Joel Cook

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

• Steel ball with steel ball
• Steel ball with glass ball

Introduction/Theory: Elastic collisions are collisions that conserve momentum and kinetic energy. This idea will be explored in this lab, as we collide two balls, in one case both made up of steel, and in another case one is steel and the other is glass. We will also trace the path of the balls after the collision. We do this because we know that the collision will conserve momentum in the x and y coordinates, and if the collision strays the two balls at certain angles, we can calculate the momentum afterwards to confirm if the momentum before and after are equal. In simpler terms, if one ball is rolling along the x-axis, and the other ball is at rest, that means that the initial momentum has no momentum along the y-axis. If the collision happens, then it is likely that there is momentum along the y-axis for both balls, but when added together, the total momentum should still be zero. This is the concept of conservation of momentum. We will also be looking at the momentum from the center of mass frame of reference to compare how it differs when we look at it from the lab frame of reference.

Procedure: We first setup the apparatus by clamping our phone with a camera on top of the apparatus. The apparatus is made up of a glass surface, a pole that extended high above the surface with a downward facing clamp for our phone. We also leveled the table, so that the ball will roll without speeding up or slowing down. Here is what it looks like:

Steel Ball with Steel Ball

We first used the balls made of steel and recorded their collision. On LoggerPro, we plotted their trajectories and graphed them on a position vs. time graph. We do not have a picture of the trajectory of the two balls during their motion, but we can show the graphs of their positions. What these graphs tells us is not only their positions but also their velocities, by looking at the slope of the lines. Here are the graphs:

The X and Y values are the coordinates for the moving ball, and the X 2 and Y 2 values are the coordinates for the ball at rest. From the graph we can see the velocities of the x and y coordinates for both balls by looking at the slope before and after the collision.

Before the collision:

After the collision:

We can now determine whether the momentum of the system is conserved after the collision. We do this by looking at the velocity of the x and y coordinates and calculating the momentum in the x and y direction separately. We measured the mass of the balls and got 67.0 g for the steel balls. We can calculate the final momentum of the balls by the summation of its masses multiplied with the final velocities, and then subtract with the summation of its masses multiplied with the initial velocities. Here is the calculation:

(Notice: the initial x and y velocities of the stationary ball is not completely zero, according to the graphs. This may be due to our imprecise plotting of the motion of the balls, or LoggerPro's estimate of the position of the ball while stationary.)

We also wanted to confirm if the kinetic energy is conserved. We converted the x and y velocities to the overall velocity by doing Pythagorean theorem, since kinetic energy is dependent on the overall velocity of the ball, not the x and y velocities separately. The calculation follows, the summation of 1/2 times their respective masses times their final velocities squared, minus the summation of 1/2 times their respective masses times their initial velocities squared. Here is the calculation for it:

As expected, the kinetic energy is conserved after the collision happened, thus confirming that this type of collision is elastic.

Afterwards, we wanted to compare our data to that of the center of mass. We made a separate graph and we then created a function that multiplied the x and y positions of both of the steel balls with their mass, all divided by the total mass, in order to get the position of the center of mass, for all of the data points. This is what the center of mass x and y graphs turned out:

X direction: Center of mass (yellow line)

Y direction: Center of mass (purple/pink line)

Upon observation, it seems that the x and y coordinates of the center of mass remains at a constant slope. In the x direction, the center of mass position is more so a constant straight line tilted slightly upward, and in the y direction the center of mass position resembles a downward line. What these graphs tells us is that the velocity of the center of mass does not change. In the center of mass reference frame, the collision is just a symmetric bounce, and the collision does not affect the center of mass at all.

We then tried to graph the x and y velocities of the center of mass by using the same function but instead multiplying the masses by velocity instead of position for each data point. We expected straight lines, but the result became this:

The data points became scattered, and this could be due to the limitations of LoggerPro and its data graphing, as it is difficult to plot with pinpoint accuracy throughout the course of the balls' motion.

Steel Ball with Glass Ball

For this next part, we replaced one of the steel balls with a glass ball and ran the same procedure as we did for the first half of the lab. Here are the results, with X and Y being the coordinates for the steel ball, and X 2 and Y 2 being the coordinates for the glass ball:

Before the collision:

After the collision:

We measured the glass ball, and its mass came out to be 19.8 grams. Here are the calculations for the conservation of momentum and kinetic energy:

(Note: there is a slightly significant error that appeared in the calculations for the y direction. The result is too high.)

After confirming that the momentum and kinetic energy are conserved, thereby making this collision elastic, we analyzed the center of mass. We did the same procedure as the first part, and here are the results:

Center of Mass X and Y Positions

Center of Mass X and Y Velocities

As expected for the center of mass positions, its slopes remained constant, thereby noting that the collision does not affect the center of mass. It is also expected that the velocity graphs of the center of mass is scattered, due to our imprecise plotting.

Conclusion: Some possible sources of errors for this lab may be from our imprecise plotting of the balls' movements, the table not being ideally leveled, and the collision losing a bit of energy in the form of vibrations, heat, and sound. Overall, the experiment showcases the concept of conservation of momentum and conservation of kinetic energy in elastic collisions. It also shows the concept that the collision does not affect the properties of the center of mass, for instance its velocity, and in the frame of reference of the center of mass, the two balls are bouncing off symmetrically.