I have a question that I don't know the answer to. Maybe one of you space cowboys can answer it for me. NASA say,"Once in Earth orbit, the last stage of the Saturn rocket fires, boosting the Apollo spacecraft and its three-man crew into its flight trajectory to the moon." Okay they fire the last stage of a big rocket to put them into flight trajectory to the moon. My question is, how come when they leave the moon's orbit they don't need to fire a big rocket to put them into flight trajectory to the earth.
They do need a
big rocket, just not nearly as big as the one that got them there.
It doesn't have to be as big for two reasons:
1. It takes less thrust to accelerate a given mass away from the Moon than from the Earth because the Moon is
much less massive.
2. The CSM's engine doesn't have to accelerate nearly as much mass as the S-IVB had to. At TLI, the S-IVB's J-2 engine had to lift the Command Module with three astronauts and provisions for more than a week, the fully-fueled Service Module, and the fully-fueled LM with its tools, fuel, and supplies
and the S-IVB stage itself with a partial fuel load. When leaving lunar orbit, the Service Propulsion System (the CSM's AJ-10 engine) had to lift the Command Module with three astronauts and provisions for three days plus a few hundred pounds of samples, and the Service Module with its partially-depleted fuel supply (probably about half at the start of the insertion burn, but that's a guess). Everything else was left behind or consumed.
NASA says,"Upon return to Earth, the command and service modules separate, leaving the command module to plunge into the Earth's atmosphere at a velocity of 25,000 mph." I've looked at the command and service module and I don't see anything that looks like the last stage of a Saturn rocket.

It may be small compared to a J-2, but that's still a pretty honkin' big rocket compared to the size of the craft it's bolted to.
Wouldn't they need the same power to get to earth as it does to get to the moon?
Nope. See above.
According to Wikipedia, the SM's AJ10-137 engine has about 9% as much thrust as the S-IVB's J-2 engine (91 kN vs 1001 kN).
Another thing, Look a the command module, how does that thing enter the earth's atmosphere perfectly with the shield pointing down and not tumbling every which way and not burning up?
Careful design, planning, and execution. They had very good rocket scientists and aerospace engineers working on it, and a well-trained crew aboard.
[Edit] Oops.. forgot the S-IVB in the list of things the S-IVB had to move but the SPS didn't.