D-1 ONE-DIMENSIONAL COLLISIONS ON AN AIR TRACK

1. PURPOSE:

To study conservation of energy and momentum in both elastic and inelatic one dimensional collisions.

2. APPARATUS:

Air track, gliders, shop vacuum cleaner or other compressed-air source, connecting air hose. Daedelon sonic ranger system, computer, software for data acquisition, computer printer.

3. BACKGROUND:

1. General Instructions: Air Tracks and Air Tables.
2. Any textbook discussion of elastic and inelastic collisions.

4. GENERAL METHODS:

Elastic collisions are achieved with spring steel bumpers on the gliders. The use of rubber or cork bumpers gives partially elastic collisions. Completely inelastic collisions require that the gliders stick together and do not separate after collision. Strips of Velcro provide an easy way to achieve totally inelastic collisions.

Elastic collisions may also be achieved by using magnets on the gliders, oriented so that approaching gliders repel each other. The gliders do not make actual contact during this kind of "collision."

Preliminary tests should be performed to ensure that:

(a) The track is perfectly level over its entire length.

(b) The air pressure is sufficient to ensure the gliders never touch the track.

One should also check:

(a) How much change in speed occurs as the glider moves the entire length of the track.

(b) How much energy and momentum are lost in an elastic collision with the bumper at the end of the track.

5. THE SONIC RANGER SYSTEM:

The sonic ranger tracks the gliders with ultrasonic pulses, measuring the time for a pulse to travel from the transducer to the reflector on a puck and back to the transducer. The reflectors on the pucks are lightweight plastic cones with interior angle 45°, so that any sound reflected from the inside of the cone travels directly back along the path it came from. This is called a retro-reflector. The air track has a transducer at each end, and may therefore track two gliders simultaneously.

The software saves the sonic data, showing it on the computer screen during the data run. The data may then be re-displayed with user-selected axis scales. You will save yourself a lot of work if you choose scales so that the marked divisions are multiples of five.

The re-scaled data may be printed, and/or saved to a computer file.

6. THEORY:

Momentum is conserved in all collisions. A perfectly elastic collision is one in which kinetic energy is conserved, that is, no mechanical energy is converted to thermal energy. A perfectly inelastic collision is one in which the maximum amount of thermal energy is produced (consistent with momentum conservation); this occurs when the colliding bodies stick or latch to each other and move off together with the same velocity.

A useful parameter for describing collisions is the coefficient of restitution γ, defined by

[1]

v' - v' 1 2 γ = - ——————— v - v 1 2

Primed velocities are the velocities after collision, unprimed ones are before collision. The subscripts label the two bodies. g = 1 for a perfectly elastic collision; g = 0 for a perfectly inelastic collision. Contact collisions for them produce deformations of the bodies, and always some loss of heat, so g is less than one. For example, g = 0.98 for a low velocity collision between hardened steel spheres. Because the amount of deformation depends on the relative impact velocity, the coefficient of restitution may also depend somewhat on this velocity.

7. PROCEDURE:

Cases to consider:

1. Elastic collisions between gliders of equal mass: (a) One glider initially at rest. (b) Both gliders initially moving.

2. Elastic collisions between unequal gliders: (a) Smaller glider initially at rest. (b) Larger glider initially at rest. (c) Both gliders initially moving, smaller glider overtakes large one. (d) Both gliders initially moving, larger glider overtakes small one.

3. Partially elastic collisions, similar initial conditions to above.

4. Completely elastic collisions, similar initial conditions to above.

8. ANALYSIS:

The Daedelon sonic ranger system software allows data sets to be saved, plotted on the computer screen, or printed. The printed plot shows the positions of both gliders as a function of time, before and after the collision. The velocities may be obtained from the slopes of the position-time plots.

Obviously you must know the mass of each glider, which you find by weighing them. You must label which glider is which on the graphs.

One objective of this experiment is to see how well your data supports the conservation of momentum. Do not simply compare system momentum before and after the collision. Compare the change of momentum of the two gliders, which should be equal and opposite. Due to experimental uncertainties they won't be, so determine the percent difference. Remember that momentum is a vector quantity, and therefore you must get the signs correct.

Likewise, compare the change in kinetic energy of the entire system due to the collision. Compare this to the initial kinetic energy of the system. This may be considerable in the inelastic collisions. The totally inelastic collision is the one in which the maximum amount of kinetic energy is lost and converted to other forms (heat, sound).

9. QUESTIONS:

1. From your data, determine whether the percent energy loss in a collision depends on the initial glider masses and velocities, or only on the nature of the impacting surfaces.

2. Why is it meaningless to use the percent change in initial and final momentum of the system in order to express how well momentum was conserved? Give an example for which such a calculation would be absurd.

3. Why is it meaningless to compare the change in kinetic energy of one glider with the change in kinetic energy of the other glider to express how well kinetic energy was conserved in a perfectly elastic collision. Give an example for which such a calculation would be absurd.

4. The position of the center of mass of the system is given by x = (m₁x₁ + m₂x₂)/(m₁ + m₂). (1) Show, from physical theory, that the velocity of the center of mass of a closed system should not change for any type of collision within the system. (2) Show, from, your data, that the center of mass velocity doesn't change in any type of collision within the system..

5. Prove, mathematically, that the retro-reflector of angle 45° reflects rays back in the direction they came, no matter what their incident direction.

10. RESULTS AND CONCLUSIONS:

How well is momentum conserved in elastic and inelastic collisions? How well is energy conserved in elastic and inelastic collisions? Does the coefficient of restitution relate to the kinetic energy loss in the collision? What fraction of the energy is lost in perfectly inelastic collisions? In all cases, be specific and quantitative.