Lorentz Contraction in Supersonic Jets

[Benzoate]

Homework Statement

Supersonic jets achieve maximum speeds of about (3 *10^-6)*c.

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

Homework Equations

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

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 .

 

[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.

 

[Benzoate]

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.

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?

 

[neutrino]

is there any other computational tool I can used that will make more precise approximations?

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.

 

[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:

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

 

[rootX]

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?

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

I got
1.00000000000225
and more precisely
1.00000000000224997798170928355350144490508802983505104210845...