Which Averages Determine the Linear Least Squares Fit in Physics Experiments?

[myelevatorbeat]

I am currently working on a lab report for my physics class. During the lab, we used air tracks, gliders, and a photogate to measure the value of 'g'. Basically, we would raise one end of the air track to a certain height and let the glider slide down the frictionless track and the timer would go on when the glider entered the photogate and the timer would end when the glider left the photogate.

Now, I'm working on the lab report and it wants to know what four averages I would use to find the linear least squares fit. I recorded the original height of the air track, the change in heights, the time it took for the glider to go through the photogate, and the instantaneous velocity when the glider was halfway through the photogate.

Can someone help me out and tell me which four values I would need to average to find the least squares fit? I know what to do from there...but I'm having a bit of trouble getting started. Here is more information, just in case:

Also, if I wanted to re-arrange this equation: mgh=1/2mv^2, how would I do it so it's a linear equation in the form y=mx+b? I figured I would solve for v^2=2gh...but then what is 'b'?

 

[Dale]

Your model is not linear in your measured values. For example, if you have a particular error in your measurement of v then that is very different from the error in your measurement of v^2. Your model uses v^2, so it is of the form y^2=mx+b not of the form y=mx+b.

However, I assume that your lab is not worried about statistical niceties like that, so that is not your real question. To answer what I assume is your real question I would say the following: In your model there is no "b", or equivalently b=0. So, if you get a significantly non-zero estimate for b in your least squares (I think you can use a simple F test for that) then you know that there is some significant problem with either your data or your model. Alternatively, many linear least squares algorithms allow you to require b=0. However, if you do that then the r value is no longer really meaningful.

I would recommend the following, run the fit normally, see if your b value is significantly different from zero. If not then re-run the fit using the option to force b to zero. Do not report r.

 

[sir-pinski]

The four averages you would need to use in order to find the linear least squares fit are the average of the original height of the air track, the average of the change in heights, the average of the time it took for the glider to go through the photogate, and the average of the instantaneous velocity when the glider was halfway through the photogate. This will give you the data points needed to plot a linear regression line and determine the slope (m) and y-intercept (b) of the line.

To rearrange the equation mgh=1/2mv^2 into the form y=mx+b, you can start by dividing both sides by m to get gh=1/2v^2. Then, you can take the square root of both sides to get √(gh)=√(1/2v^2). Simplifying further, you get √(gh)=v/√2. Finally, you can rearrange the equation to get v=√(2gh). In this form, the equation is now in the form y=mx+b, where y is the velocity, x is the height, m is √(2g) and b is 0. Therefore, the y-intercept (b) in this case is equal to 0.