Figure 37-16 shows a ship (attached to reference frame ${\mathit{S}}^{\mathbf{\text{'}}}$) passing us (standing in reference frame$\mathit{S}$). A proton is fired at nearly the speed of light along the length of the ship, from the front to the rear. (a) Is the spatial separation $\mathbf{∆}{\mathit{x}}^{\mathbf{\text{'}}}$between the point at which the proton is fired and the point at which it hits the ship’s rear wall a positive or negative quantity? (b) Is the temporal separation $\mathbf{∆}{\mathit{t}}^{\mathbf{\text{'}}}$between those events a positive or negative quantity?

The proton is fired nearly speed of the light along the length of the ship.

(a)

The spatial separation is defined as the separation between the two events is greater than speed of light, c times their time separation.

Since the given figure it can be seen that the spatial separation if directed toward the negative x axis, therefore, spatial separation is the negative quantity.

Hence the spatial separation$∆x\text{'}$ between the point at which the proton is fired and the point at which it hits the ship’s rear wall a positive quantity.

(b)

When two observers move relative to each other and measure the time with respect to each other then measurement of the time would be different for both the observer and this time measurement is known as temporal separation.

Since the temporal separation is measure the time, therefore it is the positive quantity.

Hence the temporal separation$∆t\text{'}$ between those events is a positive.