In the earth’s reference frame, a tree is at the origin and a pole is at $x=30$km. Lightning strikes both the tree and the pole at $t=10\mu s$. The lightning strikes are observed by a rocket traveling in the $x$-direction at $0.50c$.

a. What are the spacetime coordinates for these two events in the rocket’s reference frame?

b. Are the events simultaneous in the rocket’s frame? If not, which occurs first?

We have given that in the earth’s reference frame, a tree is at the origin and a pole is at $x=30$km.

Lightning strikes both the tree and the pole at $t=10\mu s$.

The lightning strikes are observed by a rocket traveling in the $x$-direction at $0.50c$.

We have to find the spacetime coordinates for these two events in the rocket’s reference frame.

$S$: Ground frame.

$S\text{'}$: Rocker's frame.

In relation to $S$, $S\text{'}$moves at a velocity of$v=0.5c$.

We have:

$$\begin{gathered} y={\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{-1/2} \\ y={\left[1-{(0.5)}^2\right]}^{-1/2} \\ y=1.15s \end{gathered}$$

The Lorentz transformations were applied to the lightning strike at $x=0m$and $t=10\mu s$,

$$\begin{gathered} x\text{'}=y(x-vt) \\ =(1.155)\left[0m-(0.5)(3\times {10}^{8}m/s)(1\times {10}^{-5}s)\right] \\ =-1732m \\ t\text{'}=y\left(t-\frac{v_x}{c^2}\right) \\ =(1.155)(1\times {10}^{-5}s-0s) \\ =11.55\mu s\approx 12\mu s \end{gathered}$$

At $x=30$km and $t=10\mu s$, for lightning strike.

$$\begin{gathered} x\text{'}=(1.155)\left[3\times {10}^{4}m-(0.5)(3\times {10}^{8}m/s)(1\times {10}^{-5}s)\right] \\ =32.9m\approx 33m \\ t\text{'}=(1.155)\left[(1\times {10}^{-5}s)-\frac{(0.5)(3\times {10}^{8}m/s)(3\times {10}^{4}m)}{{(3\times {10}^{8}m/s)}^2}\right] \\ =-46.2\mu s\approx -46\mu s \end{gathered}$$

We have given that in the earth’s reference frame, a tree is at the origin and a pole is at $x=30km$.

Lightning strikes both the tree and the pole at $t=10\mu s$.

The lightning strikes are observed by a rocket traveling in the $x$-direction at $0.50c$.

We have to find the events are simultaneous in the rocket’s frame or not and if not then which occurs first.

The events in the frame of the rocket are not all happening at the same time. The lightning strikes the pole$46+12=58\mu s$ before the free.