Lab 15: Collisions in Two Dimensions
Chris Ceron, Amy, John
Goal of Lab
Look at a two-dimensional collision and determine if momentum and energy are conserved.
Theory/Introduction
For our collision, we can split up velocity and momentum into x and y components depending on where our axis is located, and in which direction the objects move before and after a collision:
Before:
|
After:
| |||
x
|
y
|
x
|
y
| |
m1
|
m1v1ix
|
m1v1iy
|
m1v1fx
|
m1v1fy
|
m2
|
m2v2ix
|
m2v2iy
|
m2v2fx
|
m2v2fy
|
Total
|
m1v1ix+ m2v2ix
|
m1v1iy+ m2v2iy
|
m1v1fx+ m2v2fx
|
m1v1fy+ m2v2fy
|
Using the table above, we can calculate the total momentum using our x and y components:
Testing the conservation of Kinetic is much simpler. We look at the Kinetic Energy of both objects before and after a collision to ensure they are equal. In our lab, the second ball is initially at rest. The formula for conservation of Kinetic Energy becomes:
In our apparatus, we set the y-axis to follow the trajectory of moving ball before the collision. This allows for the center of mass in the x direction to equal 0, and the center of mass in the y direction to increase as the ball continues to move up the y-axis. We can calculate the center of mass using the formulas:
Apparatus/Procedure
Our apparatus is composed of a leveled, frictionless glass surface with a stand that allows for a camera to be placed directly on top of the glass. We chose to use Amy's iPhone as it has the option to record and play a video in slow motion.
For the center of mass graphs, we used the formulas stated in the Theory/Introduction and created a new calculated column for those values. Using the columns, we could create a graph of center of mass vs. time.
Graphs for our first scenario (small marble colliding into an equally sized marble)
Center of mass graphs:
Momentum broken down into x and y components:
Initial Kinetic Energy:
Final Kinetic Energy:
Graphs for our second scenario (big marble colliding into a small marble)
For this portion of the lab, we did not get the initial velocity of the big marble before the collision. We used a similar fit (X2) as the initial velocity.
Velocities of both marbles broken down into x and y components:
Center of mass graphs:
Initial Kinetic Energy:
Final Kinetic Energy:
Conclusion
The goal of the lab is to use a two-dimensional collision to determine if momentum and energy are conserved. Based on our calculations and data, we see that neither are conserved. This is due to several uncertainties found in our lab. For starters, we assumed that all of the initial energy and momentum is transferred after a collision. We did not take heat and friction during the collision into consideration. Our frictionless glass surface is also not entirely frictionless. The conservation of energy is based on no external forces interaction with the system; however, the friction found on the glass surface and on the collision of the two marbles interact with our apparatus. In our calculations of momentum and energy, we can see that energy is lost because our values before the collision are larger than the values after the collision.
Testing the conservation of Kinetic is much simpler. We look at the Kinetic Energy of both objects before and after a collision to ensure they are equal. In our lab, the second ball is initially at rest. The formula for conservation of Kinetic Energy becomes:
In our apparatus, we set the y-axis to follow the trajectory of moving ball before the collision. This allows for the center of mass in the x direction to equal 0, and the center of mass in the y direction to increase as the ball continues to move up the y-axis. We can calculate the center of mass using the formulas:
Apparatus/Procedure
Our apparatus is composed of a leveled, frictionless glass surface with a stand that allows for a camera to be placed directly on top of the glass. We chose to use Amy's iPhone as it has the option to record and play a video in slow motion.
A high frame rate allows for us to clearly see the balls in slow motion, and plot the ball's position vs. time clearly through LoggerPro. We recorded two different scenarios for this lab: a small marble colliding into a marble of equal size, and a big marble colliding into marble of smaller size.
Once we leveled the glass surface and placed Amy's phone on the stand, we recorded both scenarios and ensures that both marbles had velocities in the x and y direction after the collision. Once the videos were recorded, we imported them into LoggerPro to plot the movement of each ball relative to time. LoggerPro allows for us to rotate our axes, so we moved the y axis to follow the direction of the moving ball before the collision.
To find the initial and final velocities of each marble, we could take the slope of each line on the position graph before and after the collision.
For the center of mass graphs, we used the formulas stated in the Theory/Introduction and created a new calculated column for those values. Using the columns, we could create a graph of center of mass vs. time.
Data and Calculations
Mass of Small Marble = 0.0048 kg
Mass of Big Marble = 0.0193 kg
 |
| Velocity of the moving marble (marble 1) before collision |
 |
| Velocity of marble 1 after collision |
 | ||||||||||||||||||||
| Velocity of marble 2 after collision (was at rest before collision)
Velocities of both marbles broken down into x and y components:
|
 |
| Center of mass for x and y |
 |
| Center of mass velocities |
Before:
|
After:
| |||
x
|
y
|
x
|
y
| |
m1
|
0
|
0.0019
|
0.0007
|
0.0007
|
m2
|
0
|
0
|
-0.0007
|
0.0007
|
Total
|
0
|
0.0019
|
0
|
0.0014
|
Initial Kinetic Energy:
Final Kinetic Energy:
Graphs for our second scenario (big marble colliding into a small marble)
 |
| Velocity of X2 after collision |
 |
| Velocity of Y2 after collision |
 |
| Velocity of big marble after collision |
Velocities of both marbles broken down into x and y components:
Before:
|
After:
| |||
x
|
y
|
x
|
y
| |
v1
|
0.05646
|
0
|
0.04083
|
0
|
v2
|
0
|
0
|
0.05646
|
-0.0106
|
Center of mass graphs:
 |
| Center of mass for x and y |
 | |
Momentum broken down into x and y components:
|
Before:
|
After:
| |||
x
|
y
|
x
|
y
| |
m1
|
0.0011
|
0
|
0.0008
|
0
|
m2
|
0
|
0
|
0.0003
|
-0.0001
|
Total
|
0.0011
|
0
|
0.0011
|
-0.0001
|
Initial Kinetic Energy:
Final Kinetic Energy:
Conclusion
The goal of the lab is to use a two-dimensional collision to determine if momentum and energy are conserved. Based on our calculations and data, we see that neither are conserved. This is due to several uncertainties found in our lab. For starters, we assumed that all of the initial energy and momentum is transferred after a collision. We did not take heat and friction during the collision into consideration. Our frictionless glass surface is also not entirely frictionless. The conservation of energy is based on no external forces interaction with the system; however, the friction found on the glass surface and on the collision of the two marbles interact with our apparatus. In our calculations of momentum and energy, we can see that energy is lost because our values before the collision are larger than the values after the collision.
No comments:
Post a Comment