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# LAB 4 - LINEAR MOMENTUM AND KINETIC ENERGY

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# LAB 4: LINEAR MOMENTUM AND KINETIC ENERGY

The object of this experiment is to investigate the conservation of linear momentum and the conservation of kinetic energy in elastic collisions.  We will study collisions between two air track gliders.

Velocity measurements are used to study a complicated mechanical event: the collision of two bodies and its effect on their motion. Motions are restricted to a single straight line, and forces other than those of mutual interaction are minimized by the use of the air track.  The object is to observe transformations of momentum and energy under various conditions of collision.

## References:

Young and Freedman, University Physics, 11th Ed. with Modern Physics, Chapter 8;

Physics 3 Laboratory Measurements.

## Basis of Experiment

The physics of one-dimensional collisions is presented in the reference text.  Referring to the "system" consisting of the two interacting bodies only, the governing principles are:

If no forces act on the system other than the ones that the parts of the system exert on each other, the system's total momentum remains constant and the total energy of the system remains constant (conservation of total momentum and energy for isolated system); this is true for both conservative and non-conservative forces.

If the forces acting between parts of the system are entirely conservative, both the mechanical energy (kinetic plus potential) and the internal energy (heat and chemical forms) of the system remain constant.  Principle (1) implies that if the internal forces are non-conservative, the changes in mechanical and internal energy will be equal in magnitude and opposite in sign (i.e. the increase in internal energy will equal the decrease in the mechanical energy). Momentum and total energy are conserved for both conservative and non-conservative forces.

In a one-dimensional system, the vector momentum reduces to an algebraic number whose sign denotes its direction according to a consistent designated convention. The energy is a scalar, as always. Since the two bodies of the system do not interact with each other except when in contact, there is no potential energy in the system at times when the objects are separated. At such times, the system's mechanical energy is its kinetic energy only.

The different possibilities to which the principles apply are designated by names given to different types of collisions:

An elastic collision is one in which the internal forces are conservative, so that both mechanical and internal energy separately stay constant throughout the collision;

An inelastic collision is one in which at least part of the interaction force is non-conservative, so that both mechanical and internal energies change. An endothermic collision is one in which mechanical energy decreases and internal energy increases, that is, mechanical is converted into internal energy (usually heat). A perfectly inelastic collision is an endothermic one in which the maximum possible amount of mechanical energy is converted into internal energy; it is a collision in which the two bodies stick together and move as a unit after the event.

Some of these types of collisions will be investigated in this experiment.

In all types of collisions, momentum is conserved. Momentum conservation for a two-body system is expressed in Eq.(1), where the m's are masses and 's are vector velocities, and the subscripts refer to bodies 1 and 2 respectively.  The primed velocities are those of the bodies after collision, and the unprimed are velocities before collision.

Momentum before collision = Momentum after collision
(1)

The conservation of kinetic energy (which is conditionally conserved) is expressed in Eq.(2):

(2) .

In these equations, the left side refers to any time before the collision of bodies 1 and 2, and the right side refers to any time after the collision. Once the system is defined for 1-dimensional motion, all velocities in one designated direction are positive numbers in the equations, and all in the reverse direction are negative numbers. The signs of the velocities are essential in Eq.(1), a vector equation. In Eq.(2), where only the squares of the velocities occur, the vector direction no longer enters -- the energy is a scalar, not a vector quantity.

## Plan of the Experiment

The low friction air track is the same as has previously been used. In order to observe collisions between unequal as well as equal masses, three gliders are needed - two of about the same mass, and one of a different mass. You are to measure glider masses at the start, and use the same gliders throughout the experiment without changing their weights.

Elastic collision of the gliders is achieved by setting them on the track with elastic bumpers facing one another.

Inelastic collisions are arranged by modifying the facing ends of the gliders with a pin and putty attachments.

The photoelectric timer as used here is operated in gate mode; it reads in milliseconds (thousandths of a second) the time during which the light beam is interrupted, from which the glider's speed in passing the beam can be directly calculated. For this experiment, each glider has been fitted with a precise 10 cm aluminum mask, painted black, and it is this mask, rather than the body of the glider, that interrupts the light beam. Thus the velocity of any of the gliders is ?=10 cm divided by the reading of the traversal time. (No correction for acceleration is needed; gliders move with constant velocity both before and after the collision. The Procedure provides for checking and ensuring this.)

Two photocell bridges, one on each side of the collision area, are used (Fig. 1). Each is connected to its own timer. Thus each timer, operating in the gate mode, starts when the light of its own photocell is interrupted and stops when the light is resumed. If the timer is not reset between two interruptions, the time interval of the second interruption (after collision) is added on to that of the previous interruption (before collision). The photo-bridges should be placed so that you can note the first reading before the number is changed by a second passage. You should not reset during the collision -- you could miss counts by trying to reset.  Practice a bit to be sure you can do the readings as required. Alternatively the PascoME-9215A timers have a memory feature that can be used to store a second glider transit time. To use this memory feature set the toggle switch to MEMORYON, a red light should go on. Press RESET. Make the two measurements. The timer will display the first measurement. Record this. Press switch to READ to get the TOTAL of the first and second measurement. Subtract the first measurement from the total to get the second measurement. Note a nice summary of the PascoME-9215A photo-gate timer operating instructions appears on the bottom of each unit.

## Procedure

First weigh the two gliders and record the weights. Weights should be measured on gliders complete with masks and whatever trimming weights have been added; no changing of masses is called for here.

Since we want the collisions to happen in the absence of all external forces, it is important that the track be perfectly horizontal and as frictionless as practical. To check the condition of your track, send one glider to your right and note the time it takes to cross each of two photocells placed one meter apart (and operated in gate mode). The difference between the two times divided by the time it took to cross the first photocell will give you the fraction of the momentum lost due to friction and gained (or lost) because of gravity.

Repeat, the above procedure, this time sending the glider to your left. Check for consistency by again sending it to the right and then left.

Before making any adjustments in the air track level, see the Supplementary Procedure Section below.

There is always a small amount of friction, which will lead to some time increase in the "downstream'' counter.  The important points are that the increases should be small, and especially that the percentage increase in a downstream counter be the same, for all practical purposes, whether you are looking at left-to-right, or right-to-left motion.

To eliminate the effect of gravity, you may need to adjust the track level in small (e.g. quarter-turn) increments using the single large screw at one end of the track.  Do not touch any other adjustment screws.   Note that there may be washers to fix the position of the track on the table.  Stay at that position, since the air track, observed over a long period, has been adjusted to be much more level than the table itself. You should be able to adjust the track so that the loss in velocity is no greater than two percent and the same (essentially) in either direction.  Readjust as necessary during the course of the experiment.

The various possible collisions, such as a heavy and a light weight glider meeting head on, a heavy hitting a light at rest, a light hitting a light at rest, all with  "springy'' bumpers (elastic), and a heavy hitting a light at rest with pin and putty bumpers (inelastic) are detailed in the Data section. More details about the Procedure, especially about achieving elastic and inelastic interactions, are given below:

## Supplementary Procedure Notes

Detailed Cautionary Notes and Explanations
A. Do Not Change Transverse Track Level
Special leveling methods have been used to correct an assembly error made on several air tracks. Because of the error, the base of the track is no longer necessarily horizontal in the transverse direction when the triangular air tube (the traveling surface for gliders) is horizontal.

The tubes have been leveled transversely; do not pay attention to obvious skewness in the crossbeam and I-beam of the tracks. Do make adjustments (as you find it necessary as described above) to the single leveling screw that levels the track along the direction of the glider motion.

B. Special Collision Hardware on the Gliders
READ C. BEFORE HANDLING GLIDERS

Each glider in this experiment has been fitted with light plastic hardware in such a way that the glider mass remains balanced and constant whether it is used in elastic or inelastic collisions.

Inelastic bumpers (sticky bumpers) have pin and putty attachments mounted on them; when two gliders collide with these bumpers mounted, they stick together, making a perfectly inelastic collision.

C. Additional Handling Care
Please do not handle the gliders by their masks.

Do not disturb the trimming weights.

Remove a glider completely from the track in order to fit or remove the inelastic bumpers.

It is highly recommended that all the elastic collisions called for be done in series, followed by inelastic collisions. This minimizes the time spent in refitting bumpers at the ends of the gliders.

D. Glider Masses
Weigh each glider with bumpers on, and the ten-centimeter mask mounted.

## Data

This is not a data sheet to be filled in - it's meant as a guide.

DATA MUST BE WRITTEN DIRECTLY INTO YOUR LAB NOTEBOOK.

OBSERVE ALL THE CAUTIONS LISTED IN THE PROCEDURE SECTION AND SUPPLEMENTARY PROCEDURE NOTES ABOVE.

1. Check that track is level in the direction parallel to the direction of motion as described in Procedure section, by comparing traversal times through photo-gates for a glider moving in each direction. When the level adjustment is complete, record the following:

Glider Moving to Right t1 =... t2=..., Glider Moving to Left  t1=... t2=..., to show that the track is level within acceptable limits as described above.

Repeat this to check the level, between collisions, and adjust the level if necessary.

2. Mass Data

Three gliders, two light, one heavy.

3. Collision Data

The kinds of collisions to be studied are listed below, with the necessary data indicated.

The arrows in column 2 show the initial direction of motion. Be sure to include the arrows in your data for the motion both before each collision and after it.

Three are elastic collisions, one is inelastic. You are welcome to try other combinations if you like - think Mercedes vs. Volkswagen!

No.   Initial Direction1      Sticky Bumpers      Initial Traversal Times            After Collision Timer Readings2            After Collision Traversal Times

 t1 t2 T1 T2 t'1 t'2 1 Off (Elastic) 2 Off 3 Off 4 On (Inelastic)

1 "." means body initially at rest.
2 Do not attempt to reset timer immediately after collision; record and , the sums of the two traversals and calculate the after-collision traversal times and .

## Calculations

For each collision, calculate the vector momentum of each body before and after the collision, and the total initial momentum and final momentum of the two-body system.

Do this for each of the three sets of data you have taken for each type of collision and for each set calculate the percent difference between initial and final momentum.

%DIFFERENCE IN MOMENTUM =

Do the same for the kinetic energy, except that you need to do this only for the data in each type of collision which shows the smallest difference between initial and final momentum.

DIFFERENCE IN K.E. =

There may be large experimental uncertainties, especially where the initial vector momentum is close to zero. Rather than carry out a detailed uncertainty calculation, you can get an approximate idea of the uncertainties from the variation in difference in momentum among the three values you have obtained for each type of collision.

## Discussion

Discuss your results for momentum and kinetic energy.  Keep in mind that we expect momentum to be conserved in every collision, while kinetic energy is conserved for elastic and not for inelastic collisions. Your discussion should include estimates of uncertainties, as indicated by the spread in your repeated (3) measurements of each type of collision, and the effects of residual friction and gravity as determined in DATA Pt.I.

Do your results confirm expectations, given the experimental uncertainties?

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