Question

In the earth's reference frame, a tree is at the origin and a pole is at x=31km. Lightning strikes both the tree and the pole at t=13?s. The lightning strikes are observed by a rocket traveling in the x-direction at 0.6 c.

What are the spacetime coordinates x? tree, x? pole for these two events in the rocket's reference frame?

What are the spacetime coordinates t? tree, t? pole for these two events in the rocket's reference frame?

Answer 1

Given:

   position of the tree : x₁ = 0 Km

position of the pole = x₂ = 31 km

Lightning strikes both the tree and the pole

at a time : t =   13 ? s = 13 x10^-6 s = 1.3 x10^-5 sec

speed of the Rocket =   v = 0.6 c

   lorentz factor : ? = 1/ ?1- v² /c²

                             = 1/ ?1- 0.6²

                              = 1.25

   space - coorinate : x'   = ? (x₁ - vt )     (for Tree)

                                      = (1.25) (0 - 0.6 x3 x10⁸ x1.3 x10^-5 )

                                      =   - 2.92Km

    space - cocorinates: x ' = ? (x₁ - vt )     (for Pole)

                                         = (1.25) (31 x10³ - 0.6 x3x 10^{8_{x 1.3 x10-5}} )

                                         = 35.82 Km

   Time co-ordinate ; t ' = ? ( t - v x /c²)     (for Tree)

                                     = (1.25)(1.3x10^-5   - 0.6 x 3x 10⁸ x 0 / c²)    

                                     = 1.625 x10^-5    sec

                                      = 16.25? sec

   Time co-ordinate : t '   = ? ( t - v x /c²)     (for pole)

                                      = (1.25) (1.3 x10^-5   - 0.6 x3x10⁸ x 31 x10³ / 9 x 10¹⁶)

                                      = [2.8 x 10^-5   - 7.467 x 10^-5 ]

                                       = - 6.125 x 10^-5    sec

                                       = - 61.25 ? sec