I anticipate that if you follow the instruction to analyze the data as you 
go, at least for sample runs, that we will need three lab sessions.

Experiment: Are momentum or kinetic energy conserved?

The objective of this experiment is to study whether or not momentum or kinetic 
energy are conserved during a collision. Your conclusions are to agree with 
your experimental measurements. (On examinations, the conservation of momentum 
or kinetic energy are to agree with my lectures, at least if you want points.)

Approach: You will take two carts, roll them at each other, and measure their 
velocities before and after the collision. Thanks to clever equipment design, 
you can do to two sorts of experiments. First, the carts can be rolled at each 
other while they make a metal on metal collision and bounce off each other. 
This might be an elastic collision. Second, the carts can be rolled at each 
other while their ends are faced with Velcro. Under reasonable circumstances, 
the carts will stick to each other and roll off as a single unit. This is an 
inelastic collision. You will then compare your measurements with the 
predictions of momentum and kinetic energy conservation results.

Calibration: in order to do the experiments, you will need to use two velocity 
sensors, one at each end of the track, each watching one cart. You should test 
if you can link two rails together to get a longer run for the two carts. It is 
probably the case that by choosing where you sit you can have one lab partner 
on each of two computers, with each lab group doing its own measurements.

Your first challenge is to make sure that the two sensors agree with each 
other. The way you do this is to put the two carts on the track attached to 
each other with Velcro, roll them down the track at several speeds in each 
direction, and record the two velocities that the two sensors report. There may 
be a fluctuation from measurements to measurement; you should calculate the 
ratio of the two velocities and calculate the root mean square spread in the 
ratio.

Systematic error: we are now introducing a new fundamental experimental science 
concept, the systematic error. Most of you are not quite old enough to remember 
clocks that are wound up by hand, and that progressively gain or lose a minute 
or a couple of minutes per day. I am resisting the temptation to bring one in 
to display it. However, my good chime clock loses about two minutes a day. This 
is a systematic error. The measurements of time are reproducible, but they are 
not quite correct. Systematic errors are entirely different from random errors.

If your two velocity measurements consistently more or less agree with each 
other, you do not have an issue with systematic error. However, if one of your 
sensors consistently reports a velocity that is, say, 5% less than the other, 
you must choose one of the sensors as being correct, and correct the velocity 
measurement of the other sensor by the percent difference. You should make sure 
that you need the same correction for carts moving in each direction. The quest 
ion ?how do I know which sensor is correct?? is invalid.  By choosing one 
sensor, you are effectively choosing a local value for the meter.

It may well be the case given that we are using digital electronics that the 
systematic error is too small to measure. However, you should look for it, 
because that is the right thing to do.

General form of the experiment: you have two carts. They each weigh something. 
There are a set of three additional weights available that you can load onto 
the carts. In addition, when you roll carts at each other, you can roll them 
very slowly, somewhat more rapidly, or somewhat more rapidly than that.  The 
two carts should not always be going at the same speed. Sometimes you should 
have the heavier one going faster. Sometimes you should have the lighter one 
going faster. Sometimes you should try to roll them at about the same speed. An 
interesting alternative which may be a bit of a nuisance to pull off with hand 
launches is to have the two carts going the same direction with the cart in 
front of moving slowly and the cart in back tailgating it. Remember, some of 
you may be teenagers, but you are not taking a stock car out on a racetrack. 
The equipment is actually reasonably sturdily built, but if it goes off the 
track this is an indicator that you were sending it down the track too fast and 
should not do this again.

In a single experimental run, you select the masses for the two carts, and you 
sent them down the track at each other. You have one velocity measurement for 
each cart before the collision and one velocity measurement for each cart after 
the collision. Corresponding to each velocity there is a momentum. Calculate 
the total momentum before and after the collision. Calculate the ratio of the 
total momentum before to the total momentum afterwards. You can also choose any 
three of the momenta, calculate the fourth momentum, and compare with your 
measurement of the fourth momentum. You now compare (a ratio seems to be the 
best choice) your calculated momentum with the measured momentum. Construct a 
table showing the four momenta, the total momentum before and afterwards, the 
four calculated choices for each of the four momenta, and the ratios of the 
measured to the calculated momenta. (Why do we use three momenta to calculate 
the fourth?  Approximately speaking, this is how the neutrino was discovered.)

Calculate the average of the ratios you just calculated. Calculate the root 
mean square spread in the ratios around the average. A reasonable figure is a 
histogram, showing how often you obtained each value of the ratio. Mark the 
average and the RMS error on the histogram in the rational way. Designing a 
good histogram plot actually requires thinking. You want enough bins that you 
can see the shape of the distribution. You want sufficiently few bins that when 
you fill them up there are reasonable number of outcomes in each bin, so the 
distribution curve looks somewhat smooth rather than like a cave floor covered 
with stalagmites.

You can also calculate the total kinetic energy of the system before and after 
the collision. Before the collision, there is only one total kinetic energy. 
After the collision there is only one total kinetic energy. Produce a table for 
each of your collisions showing the two kinetic energies. In a third column, 
calculate the ratio of the two kinetic energies. Calculate the average and the 
root mean square spread in your measurements. Is kinetic energy conserved in an 
elastic collision, or is there a systematic difference between the after and 
before kinetic energies?

The repeat experiment: now attach the Velcro to the front ends of the two 
carts. Repeat your experiments, as so closely as you can, except this time the 
two carts are supposed to stick to each other. You may find that if the carts 
are moving really fast or really slow they don?t stick. Once again, calculate 
the momentum before and afterwards, calculate the kinetic energy before and 
afterwards, and see whether there is a change in the kinetic energy and the 
momentum.

Data analysis: if you think about this, you?re going to be doing the same 
calculation a really, really large number of times. You all have access to 
Maple. As students, you are all allowed to download Mathematica, though you 
will want to use a real computer with a real memory rather than your pocket 
telephone in order to do this. Some of you also may know how to use Matlab. I 
understand that in each of these languages you can write simple programs that 
will take a series of input numbers, perform operations on them, and give you a 
nice pretty output that you can paste into a spreadsheet for the table. If you 
think about how many calculations you are going to need to do, and how long 
this will take, you will hopefully realize that it is a really, really, really 
time to learn how to write a computer program in some language if you do not 
already know.

Conclusions: separately for elastic and inelastic collisions, you should report 
the ratio of the momentum beforehand to the momentum afterwards, and the root 
mean square spread in your measurements. You should report whether or not your 
measurements are consistent with momentum conservation. If your average is more 
than two root mean square spreads away from unity, your measurements do not 
support momentum conservation.

For the kinetic energy, you should report the average ratio of the kinetic 
energy before and the kinetic energy after the collision. You should make this 
calculation separately for elastic and inelastic collisions. In an elastic 
collision, there is a quantity known as the coefficient of restitution, which 
you have just measured for your carts. The coefficient of restitution is the 
fraction of the kinetic energy that comes out, given how much kinetic energy 
went in. What coefficient of restitution did you measure? Is your measured 
coefficient of restitution significantly different from unity? (You can answer 
this by considering where your average is relative to unity and relative to the 
root mean square error in your measurements.)

On the first day of the lab, you should do at least a few measurements of 
elastic and of inelastic collisions. You should try at least some combinations 
of weights and speeds. Before the second lab session, you should analyze your 
measurements from the first lab to see how things are working. They may well 
not be working. (Ideally, after each run you stop and do the calculations to 
see what is going on. If you discover, for example, that momentum conservation 
does not work when the total momentum is really close to zero, you may want to 
do more measurements with close to zero total momentum. You may also want to do 
more measurements with way away from zero momentum rather than fixing your eye 
on one peculiar feature of the data which may be a statistical accident.

In the real world, the first time you try almost any experiment the measured 
data are junk. You don?t have your hands working well yet to run the 
instrument, there are minor errors in alignment and calibration and 
experimental planning that have to be cleaned up, and you forget to do things. 
Only after several days try, if you are lucky, will the experiment work well. I 
shall give an example from personal experience. I invented a device known as 
the homodyne coincidence spectrometer. It involves two anti-parallel laser 
beams, that have to be very accurately anti-parallel, and two detectors placed 
on opposite sides of the instrument. The detectors have to be really accurately 
lined up, too. I spent a great deal of time polishing up the alignment tools in 
the alignment procedure before I tried the experiment. So I set everything up, 
turned the spectrometer on, and went out for a walk. When I came back, I was 
pleased to see that the instrument was working with about as much efficiency as 
I ever got out of it. There was, however, a highly competent English group that 
tried to duplicate the experiment and spent I gather some months on it. They 
couldn?t get it to work. Sometimes, good hands are really important if you want 
an experiment work. Sometimes a great deal of thought in advance helps. 
Sometimes, you have to be a bit lucky.