91影视

We have given that in the earth鈥檚 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 $t=10渭s$.

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鈥檚 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 $t=10渭s$,

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

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

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

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

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

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鈥檚 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$46+12=58渭s$ before the free.