Lab 15 Collisions in two dimensions

Purpose: the purpose of this laboratory is to look at a two dimensional collision and determine if momentum and energy are conserved.

The apparatus we used consisted of a board and a camera is clamped to it and recording the collision from above.

Procedure:

For this lab we set up the camera to take a video on the computer and then we recorded two collisions . The first collision that we did was between the marble and the steel ball. We recorded the collision with the camera and used the video analysis on logger pro we marked the points of the trajectories of each ball throughout the video. We then graphed the data on a position vs time graph on logger pro. Using the position and time graph we were able to get the velocities before and after the collision for the marble and the steel ball. We repeated the procedure for the second collision which involved two steel balls of equal mass. What we are trying to show is that the momentum is the same before and after the collision in the x direction and in the y direction. Also, that the energy is the same before and after the collision. To do this we will use the following equations:

Where M1 is the mass of the steel ball and Vi is the initial velocity if the steal ball and Vf1 is the velocity final of the steel ball. M2 is the mass of the Marble in the first collision and the mass of the steel ball in the second collision and Vf2 is the final velocity of the marble in the first collision, and the steel ball in the second collision.

But for the conservation of momentum equation we will see if momentum is conserved in the x direction and the y direction separately because that is what logger pro will give for velocity. In the conservation of energy equation we are going to find the magnitudes of the velocities in the x and y directions and plug the magnitude in for the velocity initial and final.

The first collision we did was with the marble and steel ball and the trajectory was the following:

The position vs. time graph of this collision looks like:

Where X and Y are the x and y positions for the steel ball that hits the marble ball and X2, Y2 are the x and y position for the marble after the collision. To get the velocity initial we took the derivative of the X and Y part of the graph before the collision. And to get the velocity final we took the derivative of the X, Y, X2, Y2 parts of the graph after the collision.

The velocities that we got are:

X V initial : -.005486 m/s
Y V initial : .3151 m/s
X V final : .02057 m/s
Y V final : .2043 m/s
X2 V final : -.08640 m/s
Y2 V final : .2783 m/s

The mass each ball:

mass of marble M2 : 19.8 g
mass of steel ball M1 : 66.6 g

Plugging in the information above into the equations  of conservation of momentum and energy and calculating the initial and final of each :

For the second collision with the steel ball the trajectory of this collision looks like:

The position vs. time graph for this collision looks like:

 

Velocities:

X V initial : .02400 m/s
Y V initial : .5796 m/s
X V final : .2190 m/s
Y V final : .3305 m/s
X2 V final : -.1914 m/s
Y2 V final : .2070 m/s

Mass of steel balls:

M1 : 66.6 g
M2 : 66.6 g

Plugging the data into the equations we get:

The data and calculations tell us the momentum and energy are conserved because the initial and final momentum values are very close to each other. The energy is also conserved for the same reason. Although, in the second collision the values are a little bit off but this had to be because of some error. The fact that the surface is not completely frictionless could be a source of error and also the tools that we used to video could be causing a little bit or error. But we learned that momentum and energy are conserved in an inelastic collision.