Question

(a)A cosmic ray proton streaks through the lab with velocity 0.82c at an angle of 52° with the +x direction (in the xy plane of the lab). Compute the magnitude and direction of the proton's velocity when viewed from frame S' moving with β = 0.74.

(b) Suzanne observes two light pulses to be emitted from the same location, but separated in time by 3.50 µs. Mark sees the emission of the same two pulses separated in time by 8.00µs.

How fast is Mark moving relative to Suzanne?c

According to Mark, what is the separation in space of the two pulses?

m

Answer 1

The x-component of the speed is,

    u_x = (0.82c) cos52⁰            (1)

The y-component of the speed is,

    u_y = (0.82c) sin52⁰               (2)

From given problem, we have

β = v/c

   = 0.74

            v = 0.74 c

Factor

γ = 1/√[1-(v²/c²)]

     = 1.49

From Lorentz transformation,(relativistic velocity transformation),

              u_x' = (u_x-v)/[1-(vu_x/c²)] .......... (3)

              u_y' = (u_y)/(γ)[1-(vu_x/c²)]   .......... (4)

Substitute the eq (1), in eq (3), we get

   u_x' = (((0.82c) cos52⁰)-(0.74c))/[1-((0.74c)((0.82c) cos52⁰ )/c²)]

        = -0.273 c

..............................................................................

Substitute the eq (2), in eq (4), we get

   u_y' = (((0.82c) sin52⁰)/(1.49)[1-((0.74c)((0.82c) sin52⁰ )/c²)]

        = 0.831 c

.................................................

Therefore velocity u = √( u_x')² + (u_y')²

                              = 0.875 c

............................................................

Angle θ = tan^-1( u_y'/u_x')

            = -71.81 0