Find the proper length of a rod if in the laboratory frame of reference its velocity is `v=c//2`, the length `l=1.00m`, and the angle between the rod and its direction of motion is `theta=45^@`.

Explanation

In the rest frame, the coordinates of the ends of the rod in terms of proper length `l_0`
`A:(0,0,0) B: (l_0costheta_0, l_0sintheta_0, 0)`
at time t. In the laboratory frame the coordinates at time `t'` are
`A: (vt^', 0, 0), B: (l_0costheta_0srqt(1-beta^2)+vt', l_0sintheta_0,0)`
Therefore we can write,
`locstheta_0=l_0costheta_0sqrt(1-beta^2)` and `lsintheta=l_0sintheta_0`
Hence `l_0^2=(l^2)((cos^2theta+(1-beta^2)sin^2theta)/(1-beta^2))`
or, `=sqrt((1-beta^2sin^2theta)/(1-beta^2))`