Theory: That a known mass going at a known velocity, colliding with another known mass, setting it and the original mass at other different but known velocities, proves that momentum is conserved (M_a*V_ao + M_b*V_bo = M_a*V_af + M_b*V_bf). Energy will also be conserved as the original kinetic energy from the first mass collides with another mass. Kinetic energy initial will equal the kinetic energy final of the new velocities of the original mass and the one it collided with (0.5*M_a*V_ao^2 = 0.5*M_a*V_af^2 + 0.5*M_b*V_bf^2).
Procedure: Set up a flat level surface, with camera to capture the collision in video, get three marbles (two steel and one aluminum), weigh for mass for each marble, set one marble in the center and roll another marble to collide with it, do two collisions (one steel marble colliding with the other steel marble and one steel marble colliding with the aluminum marble), use the video capture program to determine the velocity of the marbles before and after collision, and graph the results. Preform calculations to confirm energy and momentum are conserved.
Apparatus
*Video screen shot with Cartesian grid and a reference "bar" to determine the velocity of the marbles
Graph of two steel marbles colliding as Position in the x, y coordinates vs time
Graph of steel marble and aluminum marble colliding as Position in the x, y coordinates vs time
*The computer can also determine the velocity (in the x,y direction) of the marbles.
*We can take the average velocity in the x,y direction of each marble, before and after, the collision as well.
Graph of two steel marbles colliding as Vax, Vay, and Vbx vs time
Results of steel marble and aluminum
*Mass of steel marble 28.1 grams
*Mass of aluminum marble 10.1 grams
*Original Velocity in the x direction of rolling steel marble = 0.5961 m/s
*Original Velocity in the y direction of rolling steel marble = 0.005765 m/s
*Original Velocity in the x direction of aluminum marble = 0 m/s
*Original Velocity in the y direction of aluminum marble = 0 m/s
*Final Velocity in the x direction of rolling steel marble = 0.5622 m/s
*Final Velocity in the y direction of rolling steel marble = -0.08335 m/s
*Final Velocity in the x direction of aluminum marble = 0.1062 m/s
*Final Velocity in the y direction of aluminum marble = 0.1825 m/s
Results of two steel marbles
*Mass of both steel marbles 28.1 grams
*Original Velocity in the x direction of rolling steel marble = 0.4817 m/s
*Original Velocity in the y direction of rolling steel marble = 0.02122 m/s
*Original Velocity in the x direction of stationary steel marble = 0 m/s
*Original Velocity in the y direction of stationary steel marble = 0 m/s
*Final Velocity in the x direction of rolling steel marble = 0.2655 m/s
*Final Velocity in the y direction of rolling steel marble = -0.1624 m/s
*Final Velocity in the x direction of stationary steel marble = 0.2020 m/s
*Final Velocity in the y direction of stationary steel marble = 0.1360 m/s
Finding actual velocity
*We must remember that actual velocity can be found using the Pythagorean theorem
*So actual velocity can be found as (Actual velocity= sqr(Vx^2 + Vy^2))
Velocities of steel marble and aluminum
*Original Velocity of rolling steel marble = 0.5961 m/s
*Original Velocity of aluminum marble = 0 m/s
*Final Velocity of rolling steel marble = 0.5683 m/s
*Final Velocity of aluminum marble = 0.2135 m/s
Velocities of two steel marbles
*Original Velocity of rolling steel marble = 0.48211 m/s
*Original Velocity of stationary steel marble= 0 m/s
*Final Velocity of rolling steel marble = 0.3112 m/s
*Final Velocity of stationary steel marble = 0.2435 m/s
Calculations of Momentum sample
*We set them equal to each other, because the amount of momentum "going in" should be the same as the momentum "going out" from the collision.
Results Momentum
*First Row Two steel
*Second Row steel and aluminum
Initial
|
Final
|
Error
|
0.0135473
|
0.015587
|
15.05673
|
0.0167504
|
0.018126
|
8.209769
|
Calculations of Energy sample
*We set them equal to each other, because the amount of energy "going in" should be the same as the energy "going out" from the collision.
Results Energy
*First Row Two steel
*Second Row steel and aluminum
Initial
|
Final
|
Error
|
0.009985
|
0.009357
|
-6.28943
|
0.0049925
|
0.004768
|
-4.49904
|
Conclusion
*It would seem the conservation of momentum of and energy has been confirmed (the numbers are close), though the amount of error is a little troubling
*Most likely because of how much rounding was required throughout the calculations, more and more rounding error was accumulated.
*Not only that but the trials don't account for energy lost due to friction(no matter how smooth the surface was) or the production of sound and heat from the collision.
*The positions of the marble before and after collision the camera used to determine velocity was manually placed, the points were placed frame by frame by where they were seen, this could have made the velocities less accurate.
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