physics question help! An observer, moving at a speed of 0.995c relative to a rod...?

Question

An observer, moving at a speed of 0.995c relative to a rod measures its length to be 2.10 m and sees its length to be oriented at 30.0° with respect to the direction of motion.
(a) What is the proper length of the rod?
b) What is the orientation angle in a reference frame moving with the rod?

Answer 1

this can be solved using Lorentz transformation

Answer 2

L = l/sqrt(1 - (v/c)^2); where L = rest length of the rod outside the moving reference frame and l = length of rod as observed from inside the moving reference frame. The denominator is the so-called Lorentz transform where v = velocity of the moving frame and c = light speed.

a) d = sqrt(l^2 + H^2); where d = 2.1 m as observed from the moving frame at v = .995c and d sin(theta) = H = the length of the rod perpendicular to the direction of travel and as seen from both inside and outside the moving frame. D = sqrt(L^2 + H^2), the proper length of the rod as seen from outside the moving observer. Since d is given and l seen by the moving observer is l = d cos(theta), where theta = 30 deg, we can solve for H. And, using L = l/sqrt(1 - (v/c)^2); we can solve for L. We then solve for D = sqrt(L^2 + H^2) to get the rest length of the rod...your "proper" length I believe.

b) The rod sitting outside the observer frame would be oriented at ARC sin(H/D) whatever that turns out to be.