A rod of propoer length l0 is at rest in frame S'. it ilies in the x', y' plane and amkes an angle of sin^-1 (3/5) What must be the value of v if as measured in S the rod is at a 45 degree)

v = 1.98*10^8 m/s

Explanation:

Given:

- Rod at rest in S' frame

- makes an angle Q = sin^-1 (3/5) in reference frame S'

- makes an angle of 45 degree in frame S

Find:

What must be the value of v if as measured in S the rod is at a 45 degree)

Solution:

- In reference frame S'

x' component = L*cos (Q)

y' component = L*sin (Q)

- Apply length contraction to convert projected S' frame lengths to S frame:

x component = L*cos (Q) / γ (Length contraction)

y component = L*sin (Q) (No motion)

- If the rod is at angle 45° to the x axis, as measured in F, then the x and y components must be equal:

L*sin (Q) = L*cos (Q) / γ

Given: γ = c / sqrt (c^2 - v^2)

c / sqrt (c^2 - v^2) = cot (Q)

1 - (v/c) ^2 = tan (Q)

v = c*sqrt (1 - tan^2 (Q))

For the case when Q = sin^-1 (3/5) ::

tan (Q) = 3/4

v = c*sqrt (1 - (3/4) ^2)

v = c*sqrt (7) / 4 = 1.98*10^8 m/s