This report verifies the theory behind collision in two dimensions which is believed to obey the laws of conservation of momentum. According to conservation of linear momentum and energy, the two are conserved in an elastic collision. However, in a perfectly inelastic collision only the former is conserved. Thus, the essence of this experiment is to verify the validity of this theory.
For this experiment, two pucks (incident and target) were used to enhance two types of collisions (perfectly inelastic and inelastic). In case of a perfectly inelastic collision, Velcro was used to bond the pucks. The experiment was computerized such that a surface-sensitive air table was used to detect the instantaneous trajectories inform of X and Y co-ordinates. Their respective velocities and angular velocities were detected and then recorded. Concurrently, graphs of relationship between the trajectories against time were plotted for the purpose of detecting the existence of angular rotation. These data were later on used in calculating the momentum; kinetic energy (K.E), fractional change in K.E and fractional changes in momentum (Deutsch 3). Consequently, a relationship between the final and initial quantities from the data were drawn, and then compared with the theoretical principle to verify its validity.
The experiment revealed that there was a fractional change in K.E and momentum for the elastic collision along the two axes with x-axis showing high discrepancies. The ranges in change in momentum were 42-49% and 31-35% along the X and Y axes respectively. The ranges in K.E were between 44-57% and 14-21% for X and Y axes respectively. This disparities could be attributed to experimental errors e.g. external forces. For the case of inelastic collision; the disparities were even greater, showing momentum change of between 44-75%. This goes contrary to our expectations theoretically hence; the data collection procedure was apparently subjected to external influence. This is the reason behind greater uncertainties in the data.
Experimental objectives
The objective of this experiment is to affirm that when bodies are involved in an elastic collision in two dimensions; just like in one dimension, both the momentum and the energy are conserved. However, this is not the case with a perfectly inelastic collision where only the momentum is conserved.
Procedure
In this experiment; a computer-analyzed collision of two pucks (incident and target of equal masses) were either set to involve in an elastic or perfectly inelastic collision on an air table. The later type of collision was achieved by the virtue of a Velcro wrapped around the pucks. Prior to launching of the pucks, the air table was leveled and an air cushion generated to minimize uncertainties due to friction forces. The degree of accuracy was further increased by launching the pucks in relatively low speeds. To compute the existence in the rotation of the pucks; the diameters of the pucks and, the relative positions of the center and a white spot at the periphery of the pucks were recorded. As regards perfectly inelastic collision after collision, the center of mass of the system (joint) and the center of one of the pucks gave the trajectories. The radius of one of the pucks was also computed.
A computer generated graph of the instantaneous trajectories along both axes against time was plotted for both incident and target pucks.
Results
The tables below represent the final results
Elastic collision
Table A of momentum before collision of elastic collision
Table B of momentum of elastic collision after collision
Table C of K.E of elastic collision after collision
Table D of the final K.E of elastic collision
Table E of fractional change in K.E and momentum under elastic collision
Table F of the momentum after collision in an inelastic collision
Table G of K.E of a perfectly inelastic motion after collision
Table H of perfectly inelastic collision showing the fractional changes in K.E and momentum
Data analysis
Elastic collision
Table1 of momentum before collision
Momentum=M1iV1i+M2i+M2i+w1i+w2i= 0.04945*(-0.541) +0.04945*(0) + 0+0 = -0.02675 NM
Table 2 of momentum of elastic collision after collision
Momentum=M1f* V1f+ M2f*V1f+w1f+w2f= 0.04945*(-0.447)+0.04945*(-0.36) = -0.03991NM
Table 3 of the initial K.E In elastic collision
K.E=0.5*M1iVi^2+0.5*M2iV2i^2+0.5*0.5*M1i*R1^2wi1^2+0.5*0.5*M2i*R2^2*w2i^2
=0.5*0.04945(-0.541) ^2+0.5*0.04945*0^2+0.5*0.5*0.04945*0.09^2*0^2+0.5*0.5*0.04945*0.9^2*0^2
= 0.007237J
Table 4 of the final K.E of elastic collision
K.E=0.5*M1fVf^2+0.5*M2fV2f^2+0.5*0.5*M1f*R1^2wf1^2+0.5*0.5*M2f*R2^2*wf2^2
=0.5*0.04945(-0.477) ^2+0.5*0.04945*(-0.36) ^2+0.5*0.5*0.04945*0.09^2*0^2+0.5*0.5*0.04945*0.9^2*0^2
= 0.011349 J
Table 5 of fractional change in K.E and momentum under elastic collision
Fractional change in momentum= (Pi-Pf)/Pi
= (-0.02675 –0.03991NM)/-0.02675=0.491682
Fractional change in K.E= (K.Ei-K.Ef)/K.Ei
= (0.007237-0.011349)/ 0.007237=0.568291
Perfectly Inelastic collision
Table 6 of the momentum after collision in an inelastic collision
Momentum= (M1+M2)*Vfc^2+3(M1+M2)*Rc^2
= (0.04945+0.04945)*(-0.447)^2+3(0.04945+0.04945)*0.09^2
=-0.04181NM ≈ 0.04181NM
Table 7 of K.E of a perfectly inelastic motion after collision
K.E=0.5*(M1+M2)*Vfc^2+3/2(M1+M2)*Rc^2*wf^2
= (0.05178+0.0515)*(-0.447)^2+1.5(0.05178+0.0515)*0.18^2*0.003417^2
=0.056624J
Table 8 of the angular velocity
Angular velocity= ((Vx-Vcenter x) ^2+ (Vy-Vcenter y)^2)^0.5/R
= ((0.005067-0.004542) ^2+ (0.00731-0.00428)^2)^0.5/0.09
=0.034168 rad/s
Table 9 showing the fractional changes in K.E and momentum
Fractional change in K.E= (K.Ei-K.Ef)/K.Ei
= (0.007237-0.056624J)/ 0.007237=-6.824263
Fractional change in momentum= (Pi-Pf)/Pi
= (-0.02675 –0.04181)/-0.02675=-0.562805
Discussion
The essence of carrying out this experiment was to verify the theory behind elastic and perfectly inelastic collisions in two dimensions. It is believed that just like in one dimension, a two dimension collision obeys the theory of conservation of momentum irrespective of the type of collision. However, this is not the case with energy conservation where only elastic collision conserves the energy before and after collision. It therefore follows that in both elastic and perfectly inelastic collisions; the fractional change in momentum (∆p) ought to be zero (Dexler 24). Since 100% conservation is unrealistic, a less than 10% uncertainty is allowed. This degree of tolerance covers the energy loses due to sound and friction. With respect to elastic collision, the fractional change in momentum showed discrepancies of between 42-49% and 31-35% for the X and Y axes respectively. Even though 100% preciseness is unattainable, greater disparities of this magnitude have a bearing on the experimental design. Some of the systematic errors that might have occurred are: the air cushioning might have influenced the speeds of the pucks; there was probably insufficient caution on the perfect conditions to carry the experiment. This experiment ought to have been carried out in a closed room to minimize the influence of wind which might slow or accelerate the speeds of pucks. Similarly, greater deviations were seen in the perfectly inelastic collision revealing uncertainties of between 44-56% and 70-76% along the X and the Y axes respectively. Correspondingly, the sources of errors have a bearing on the experimental design with greater influence skewed towards the provision of an air cushion.
With regards to fractional change in K.E, the K.E loss for an elastic collision ought to have been less than 20%. This tolerance cover losses due to friction on impact and sound energy loses. In our case, the energy loses were within the ranges meant for elastic collision as shown by the Y axis of the same. This showed a range of between 14-21% confirming the principle. However, along the X axis the energy loses ranged between 44-57%. These deviations have a bearing on the systematic errors explained initially in fractional changes in momentum.
The errors explained in this experiment can be minimized by carrying out the experiment in an area free of air currents. This can be achieved by setting out the experiment in an enclosed room with fans turned off. Otherwise, the errors would recur. The pressure gauge of the air flowing towards the air table ought to be checked and confirmed that is flowing at a constant rate to minimize uncertainties.
Conclusion
The objective of the experiment was to affirm the principles of conservation of momentum and energy in two dimensions for both elastic and perfectly inelastic collisions. The experimental data revealed a correlation between the two in line with the theory though to a certain degree. As much as the degree of preciseness was low, the momentum was conserved at above 60% for elastic collision in all cases. Perfectly inelastic collision recorded a minimal momentum conservation of less than 50% in all cases. The kinetic energy loses were within the expected ranges of less than 20% for the elastic collision along the Y-axis. However, the X-axis recorded losses of more than 20 but less than 44%.
Works cited
Drexler, Jerome. How Dark Matter Created Dark Energy and the Sun: An Astrophysics Detective Story. Makawao, Maui, HI: Inner Ocean Publishing, 2003. Print.
Deutsch, Oshua. Elastic Collision in two dimensions: New York: Random, 1997. Print.