An explosion

Astronomers observe the planet Mars as the Martians fight a nuclear war. The Martian bombs are so powerful that they rip the planet into three separate pieces of liquified rock, all having the same mass. If one fragment flies off with velocity components

v_1x = 0

v_1y = 1.0 × 104 km/hr ,

and the second with

v_2x = 1.0 × 104 km/hr

v_2y = 0 ,

(all in the center of mass frame) what is the magnitude of the third one's velocity?

In the center of mass frame, the planet initially had zero momentum. After the explosion, the vector sum of the momenta must still be zero. Vector addition can be done by adding components, so

mv_1x + mv_2x + mv_3x = 0 , and

mv_1y + mv_2y + mv_3y = 0 ,

where we have used the same symbol m for all the terms, because the fragments all have the same mass. The masses can be eliminated by dividing each equation by m, and we find

v_3x = -1.0 × 10⁴ km/hr

v_3y = -1.0 × 10⁴ km/hr

which gives a magnitude of

= 1.4 × 10⁴ km/hr