Question

5. (Fully graded)An observer in frame S sees lightning simultaneously strike two points 100 m apart. The first strike occurs at zi = V1 =句= t,-0 and the second at z2 = 100m, y2 = 뀨2 = t2 = 0. (a) What are the coordinates of these two events in a frame S" moving at 0.70c in the positive x direction relative to S? b) How far apart are the events in S? (c) Are the events simultaneous in S? If not, what is the difference in time between the events and which event occurs first?

Answer 1

Frame $S'$ is moving along the common X axis relative to frame $S$ with a speed $v=0.70c$.

Lorentz transformations relating are

$x' =\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}$, $y'=y$, $z'=z$ and $t'=\frac{t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}$

(a)

The two events in the $S$ frame are $(x_1,y_1,z_1,t_1)=(0,0,0,0)$ and $(x_2,y_2,z_2,t_2)=(100\,m,0,0,0)$

$x'_1 =\frac{x_1-vt_1}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{x_1-0.70ct_1}{\sqrt{1-0.70^2}}=\frac{0-0.70c*0}{\sqrt{1-0.70^2}}=0\,m$

$y'_1=y_1=0$

$z'_1=z_1=0$

$t'_1=\frac{t_1-\frac{vx_1}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{0-\frac{0.70c*0}{c^2}}{\sqrt{1-0.70^2}}=0\,s$

The two events in the $S'$ frame are

$(x'_1,y'_1,z'_1,t'_1)=(0\,m,0,0,0\,s)$

$x'_2 =\frac{x_2-vt_2}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{x_2-0.70ct_2}{\sqrt{1-0.70^2}}=\frac{100-0.70c*0}{\sqrt{1-0.70^2}}=140.028\,m$

$y'_2=y_2=0$

$z'_2=z_2=0$

$t'_2=\frac{t_2-\frac{vx_2}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{0-\frac{0.70c*100}{c^2}}{\sqrt{1-0.70^2}}=-\frac{98.019606}{c}=-3.269582*10^{-7}\,s$

$(x'_2,y'_2,z'_2,t'_2)=(140.028\,m,0,0,-3.269582*10^{-7}\,s)$

(b)

The events are $x'_2-x'_1=140.028-0=140.028\,m$ apart

(c)

The events are not simultaneous.

$t'_1=0\,s$

$t'_2=-3.269582*10^{-7}\,s$

The time difference is $t'_1-t'_2=3.269582*10^{-7}\,s$

The second event (with subscript 2) occurred first.