Mary, the commander of the spaceship just discussed, is holding target practice for junior officers by shooting protons at small asteroids and space debris off to the side (perpendicular to the direction of spaceship motion) as the spaceship passes by. What speed will an observer in the space station measure for these protons?
Strategy We use the coordinate systems and speeds of the spaceship and proton gun as described previously. Let the direction of the protons now be perpendicular to the direction of the spaceship—along the y’ direction. We already know in the spaceship’s K’ system that u_{y}^{\prime}=0.99 c \text { and } u_{x}^{\prime}=u_{z}^{\prime}=0 and that the speed of the K’ system (spaceship) with respect to the space station is v 0.60c. We use Equations (2.23) to determine u_{x}, u_{y}, \text { and } u_{z} and finally the speed u.
u_{x}=\frac{d x}{d t}=\frac{\gamma\left(d x^{\prime}+v d t^{\prime}\right)}{\gamma\left[d t^{\prime}+\left(v / c^{2}\right) d x^{\prime}\right]}=\frac{u_{x}^{\prime}+v}{1+\left(v / c^{2}\right) u_{x}^{\prime}} (2.23a)
u_{y}=\frac{u_{y}^{\prime}}{\gamma\left[1+\left(v / c^{2}\right) u_{x}^{\prime}\right]} (2.23b)
