Final answer:
The explosion of an oxygen tank in the service module while in space could result in the loss of cabin pressure, depletion of life support, and potential damage to critical spacecraft systems, while the spacecraft's center of mass would continue to travel uniformly per the conservation of momentum principle.
Step-by-step explanation:
The effect of one of the oxygen tanks in the service module exploding in space could be catastrophic for any space mission. This event would likely lead to a rapid loss of cabin pressure, deplete vital life support reserves, and could potentially damage other systems critical to the spacecraft's functioning. In the vacuum of space, the explosion would disperse debris in all directions, potentially posing risks for spacewalks or damage to the spacecraft's exterior, which could impact thermal regulation among other things. There would be no sound associated with the explosion as there is no air in space to carry sound waves. However, astronauts inside the spacecraft might hear sounds of the explosion transmitted through the structure of the spacecraft itself.
Regarding the motion of the center of mass after such an explosion, the conservation of momentum dictates that, if no external forces act on the system, the center of mass would continue moving uniformly, as before the explosion. This means that while the individual pieces from the explosion follow their own trajectories, the collective center of mass for the system remains constant in its path unless acted upon by external forces, like gravity or other objects. The same principle can be applied to describe an astronaut's movement in space: by expelling mass in one direction, they will propel themselves in the opposite direction, in accordance with Newton's third law of motion.