Introduction and Theory
The main aim of this experiment is to investigate energy and momentum conservation in a closed system. Momentum and energy conservation principles are significant physical variables that engineers use when describing various physical phenomenon. Momentum of a body is a vector quantity, which is given by the product of its velocity and mass. In collision theory, the momentum of the body is always conserved when two bodies collide. The principle of momentum conservation applies to both elastic and inelastic conservation irrespective of the dynamic interaction between the two objects. This report demonstrates the application of momentum equation in determination of motion parameters (velocity and kinetic energy) of two bodies after collision when the motion parameters before collision are known.
The total energy of a system comprises of the summation of the different forms of energy such as kinetic energy and potential energy. The kinetic energy is the energy associated to body due to its motion. For a single body or particle the mathematical expression for kinetic energy is:
k=12mv2.1,
where, m and v respectively represent the mass and the velocity of the body. Conservation of the total energy of the system is another fundamental conservation law in physics.
Elastic collision
In an elastic collision, bodies rebound or bounces from each other after collision. The kinetic energy and momentum of the system is thus conserved in an elastic collision. The mathematic expressions involved are described below:
The following mathematical notations have been applied throughout this report:
m1 = mass of cart 1.
m12 = mass of cart 2.
v1i= Initial velocity of cart 1
v2i= Initial velocity of cart 2
v1f= final velocity of cart 1
v2f= final velocity of cart 2
The conservation of momentum for a closed system is expressed as shown below:
pi= pf (2)
Initial momentum
pi=m1v1i+m2v2i (3)
Final momentum
pf=m1v1f+m2v2f 4)
The initial kinetic energy of the system is
k1= 12m1v1i2+ 12m2v2i2 (5)
Final kinetic energy becomes
kf= 12m1v1f2+ 12m2v2f2 (6)
Thus, for an elastic collision the total energy of the system is conserved. Thus,
ki=kf ( 7)
The deviation between the initial value and the final values is obtained from the following mathematical expression.
% discrepancy=(Xf-Xi)Xfx 100 % (8)
Inelastic collision
A collision is said to be inelastic when the two bodies collide and sticks together or fails to rebound. The kinetic energy in this case is not conserved. However, the total momentum of the system is conserved. Experimental investigations have shown that the final kinetic energy of the system is less than the initial kinetic energy. The difference is attributed to the energy converted to heat, potential energy and the energy used in the physical deformation. However, it is important to note that the final kinetic energy may be larger than the initial one if some potential energy is released during the collision.
Momentum conservation
pi= pf (9)
Initial momentum
pi=m1v1i+m2v2i (10)
The two bodies considered in an inelastic collision herein stick together after collision and move with a common final velocity. Hence, final momentum is given by:
pf=m1+m2xvf (11)
Initial kinetic energy
k1= 12m1v1i2+ 12m2v2i2 (12)
Final kinetic energy
kf= 12m1+ m2vf2 (13)
For inelastic collision
ki>kf
m1v1i+m2v2i=m1+m2 x vf
when v2i=0, the above expression simplifies to:
m1v1i=m1+m2xvf (14)
We can define a parameter x=m1m1+m2 and use it to eliminate vf . Then applying equations 12, 13 and 14 to we can show that
% change in kinetic energy=kf-kikf*100 % (15)
Graph
The record experimental data was tabulated in Table 1 as shown:
The uncertainty of the various discrepancy values for elastic and inelastic case is demonstrated by the graphs of relative change in total momentum and kinetic energy versus the number of trial or measurement (4 measurements were done in each case). Trials 1 to 4 are for elastic collision and trials 5 to 8 are for inelastic collision the graphs are shown below:
Questions
Question 1-Comparison of the initial and final momentum for the elastic cases considered shows a huge deviation between the final and the initial values. The discrepancies are large when the initial velocity of cart 2 before collision is zero. This is because there exists some resistance to motion or impulse when the two bodies collide. This affects the motion of the body.
Question 2- The initial momentum before the collision is greater in cart 1 since the momentum of 2 is zero (v2 is zero). After collision, the velocity of the cart 2 is greater than that of cart 1. Therefore, the contribution of the second cart to the total momentum is greater in cart 2.
Question 3- For the elastic case, the kinetic energy before and after collision should be conserved. However, in this case the kinetic energy of the system is not conserved within the expected limits of accuracy. The deviation observed in this work is attributed to experimental errors.
Question 4 - we have noted that the kinetic energy and momentum in work was not conserved. We have attributed this to the fact that some of the energy before collision is changed to other forms of energy.
Question 5-From theory we have seen that the momentum of an inelastic system should be conserved. However, the result obtained in this works shows a very huge deviation between the initial momentum and final momentum. This happens because of the various errors encountered in this work. These have been discussed in the conclusion.
Question 6-For the inelastic case, the contribution of cart 1 to the total momentum before collision is greater than that of cart 2. This is because; the velocity of cart 1 is greater than that of cart 2 for all the cases involved. Contribution of cart 2 is greater only when the 100 gram mass is not added to cart 1. This is because after collision, the two bodies stick together and have the same velocity.
Question 7- For the inelastic case, the experiment confirmed that the kinetic energy is not conserved. The large deviation between the initial values and final values evidences the fact that kinetic energy is not conserved.
Question 8- It is impossible to have a perfect elastic collision case in real life situation. This is because kinetic energy will always be converted to other forms of energy such as sound and heat or absorbed in deformation as long as two bodies collide.
Question 8-In this experiment, it is very clear that, the error margin in both elastic and inelastic collision is bigger when the initial velocity of cart 2 is zero. We have also observed the change in direction of cart1 after collision when the two carts were moving before the collision.
Conclusions
This work has demonstrated the use of the elastic and inelastic theory to prove the various conservation principles. The initial parameters before the collision were specified, while the conservation principles were used to compute the final parameters after collision. From this work, the discrepancy between the experimental values and expected values is very large. This emanates from the various errors encountered during experimental set-up and data analysis. The various errors encountered are listed below:
1. The observed vibration of the carts and tracks during the experiment largely affected the results obtained in this work. The vibration slowed the carts resulting to lower velocities.
2. The analysis involved herein assumes constant velocity of the carts. That is the final velocity of the carts as they pass through the gates was considered. The effects of deceleration and acceleration of the tracts were thus ignored.
3. Finally, the tracks were assumed to be perfectly straight. Misalignment of the tracks could have caused some errors in the experimental values for momentum obtained.
These were the three major sources of errors observed in this experiment. Thus, to reduce these errors the following should be ensured.
1. The straightness and the level of the tracks should be checked and corrected for any misalignment.
2. The carts should be pushed slowly during the experiment to reduce the effects of vibrations
3. The mathematical expressions should be modified to include the effect of acceleration of the carts or bodied before and after collision. The instantaneous velocity of the cart or the object should be captured. This would give a very accurate value of the momentum and kinetic energy.
