Lorentz contraction problem.

1. Aug 13, 2007

Benzoate

1. The problem statement, all variables and given/known data
Supersonic jets achieve maximum speeds of about (3 *10^-6)*c.

By what percentage would observe such a jet to be contracted in length?

2. Relevant equations

The equations for these this problem would be : 1/gamma= L(proper)/Length = 1/sqrt(1-v^2/c^2) ; beta=v/c

3. The attempt at a solution

1/sqrt(1-(v/c)^2)^.5 => 1/sqrt(1-(beta)^2)= 1/sqrt(1-(3e-6)^2)^.5=1=gamma; which leads to gamma=1=L(proper)/Length or L = (1/gamma)*L(proper) = > L=L(proper) leading me to the conclusion that the length of the jet did not contract at all .

2. Aug 13, 2007

neutrino

You must remember that you have approximated the value of gamma to 1, and that DOES NOT imply that no contraction occurs at all. A better answer would be that the contraction that does occur is too negligible to be observed.

Btw, your formula for length contrction: $$\frac{1}{\gamma} = \frac{L_o}{L}$$, is incorrect.

3. Aug 13, 2007

Benzoate

Oh okay, thank you. the correct formula should be gamma=L(P)/L My TI 83 Plus will only make approximations to 1. is there any other computational tool I can used that will make more precise approximations?

4. Aug 13, 2007

neutrino

I'm sure there must be one, but I'm afraid I don't know any. All calculators I have access to give me 1.

5. Aug 13, 2007

G01

You can try a computer algebra program, but I can't guarantee it will work. Your best bet would probably be to use the binomial approximation on gamma.
Taking only the first two terms of the approximation should be exact enough here. If not, the approximation to three terms is:

$$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2$$

Last edited: Aug 13, 2007
6. Aug 14, 2007

rootX

for this exp:
1/sqrt(1-(3e-6)^2)^.5?

I got
1.00000000000225
and more precisely
1.00000000000224997798170928355350144490508802983505104210845...