Saturday, May 17, 2014

Lab Book 2014_05_16 Fiberglass Dewar Leaks and more on Relativistic Projectile Range

Lab Book 2014_05_16     Hamilton Carter

Summary
There's a minor setback.  The fiberglass Dewar has a leak.  On the theory side of things, the relativistic range equation is shown to be proper x velocity times the rapidity of the y component of velocity divided by the acceleration due to gravity which is about what you'd expect it to look like if you first looked at the classical result and then squinted.  Work is being done to determine what, if anything, to make of the result.  Notes and a brief Mathematica file are included.

Leak testing the fiberglass Dewar today.  I’m also looking through the second Tehran paper.


There is a leak at the Teflon joint that is away from the Dewar.  It’s a rather small leak that can’t be detected using the roughing pump gauge.


I've very slightly opened the Dewar valve and begun pumping on the large volume.  After about five minutes, the vacuum was back down to 3 * 10^-1.

It appears that the fiberglass Dewar seals did not reseat. I opened the valve a little more which I expected to improve the vacuum pressure if we didn't have a leak since it would increase the available pumping speed.  The pressure got worse and stayed worse.   The mechanical pump is not pulling below 9*10^-1 after about an hour of pumping.

Theory Work
I showed that Shahin and MacColl’s range equations are equivalent.  The relativistic range equations from Shahin’s paper and my extension, (from yesterday),are


and


I was wrong yesterday.  The factors to the left of the log are constants, but it’s interesting to vary beta.  When it goes to 1, the expression goes to infinity.  There’s one other interesting issue though.  The range expression can also be written as:


Written this way, it’s the horizontal component of proper velocity times the vertical component of rapidity dived by the acceleration of gravity. The product of the speed of light times the veritical component of rapidity divided by g gives a proper time that’s equal to the hang time of the projectile.  There’s a second expression for the vertical hang time agreed upon by both MacColl and Shahin.  The expression is


converting this to proper time, we get


There’s probably a mistake here somewhere, but I’m not seeing it today.

Shahin and MacColl’s range equations were compared [mathematica file] .  They’re quite different notationally, but they are mathematically equivalent.


More raw notes:











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