Electronic – What wire would I need for 24000 watts (48v @ 500amps max) of DC power

You probably want a stored energy system to reduce required motor power.
Mean power can be low - if you lower the firing rate the mean power drops.

Easiest way to get consistent results is probably to have a rotating wheel with substantial mass that spins at the required circumferential velocity. If this is used to accelerate and "sling" the puck, and if Mwheel >> Mpuck then it will not slow much as the puck is accelerated to exit speed. Wheel is spun up by a motor of whatever power is desired. Larger motor = quicker recovery time. Note that if the wheel stores say 10 x the puck max energy you have a potentially lethal flywheel which would not be too hard to build safely, but care is required.

Wheel balancing machines spin the tires & wheels up with an electric motor. Some have quite a small Wattage motor and a slow run up time. Others use larger motors and faster runup. I've seen older ones with a 24 VDC motor and maybe a few 10's of Watts max power.

Spin up a wheel on such a machine to desired speed, place a sprung "floor" under the wheel, not touching but less than puck height separation, feed in a puck [from the BACK :-) ! ] and watch it vanish as it touches the tire. An old deadish wheel balancer may be available at minimal cost. Probably larger than what you had in mind :-).

Energy & Power:

• Liberties: The following takes some liberties with power and energy profiles when accelerating a puck. If you accelerate at constant power then velocity change is not linear. If you accelerate at constant acceleration then power input is not linear. Time to accelerate depends on power input profile and is not the simplistic figure derived below. But, the following should give a good feel for power end energy levels involved.

Energy / Power / Acceleration / time in 'launcher' / ... :

Given puck with mass = 100g = 0.1 kg and
velocity_max = 60 mph = 27 m/s -> say 30 m/s then

Puck energy \$(E_K) = \frac{1}{2} \times m \times v^2 = 0.5 \times 0.1 \times 30^2 = 45 Joule = 45 Watt seconds. \$
(\$ E_K \$*=kinetic energy, m= mass of the object in kg, v= speed of object in m/s).*

Linear motor: If you accelerated it at constant power over a say 1 metre linear track them
Vmean = (30-0)/2 = 15 m/s so time to accelerate = 1/15s = 67 mS.
As puck energy = 45 W.s and you are delivering this in 1/15s the power during acceleration is ~= 45 Ws / (1/15s) = 675 Watts.
That's more Watts than you'd like to need in a linear launcher if you are driving it directly electrically.

"Crossbow": Instead you can wind up a "crossbow" type mechanism where a spring or torsion bar or whatever is wound up to store the desired amount of energy and then "tripped".
For a potential firing rate of 1/second you need 45 Watt of energy to storage transfer rate - say about double this to get electrical motor input or about 100W.
For a 5 second repeat rate you are down to 20W and 10W for 10 second cycle time.

"Putter:" A 10 Watt motor winding up a spring via a reduction box or screw thread or ...? is easy to build and play with. It would (probably) be easy to pull back a weight on a pivot (rigid shaft pendulum) and let it go so it swings and strikes the puck, golf putter style. Potential energy in a mass is mgh
or about 10 x kg x height.
You want 45 W.s max. 10 kg x 0.5 metre x 10 = 50 W.s
Height is the vertical height above rest level.
Velocities are now wrong and you are going to need to deal with impulse energy transfer, but it should be in the order of right.

Rocket: A potentially workable system would be to use compressed air and a pressure reservoir. A completely DIY system could use a "water rocket" type launcher as built by amateurs world wide. Liable to be somewhat noisy [tm] on release.

Treadmill motor / Direct drive rotary flinger: A potentially excellent excellent motor for a flinger is a treadmill motor. These are typically rate in the 0.5 to "several" HP range and 200 VDC permanent magnet units powered from 230 VAC rectified mains (and no doubt the 110 VAC based versions in the US) are common. Speed control by PWM or other voltage variation allowing wide range speed control. An arbitrarily large disk could be attached directly to suit the motors preferred RPM maximum. eg an 1800 RPM motor = 30 RPS will need a 1 metre circumference disk to deliver 30 m/S = a diameter of 318 mm or about 1 foot.

I once visited a hackerspace that was trying to make a light bulb throb between bright and dim for a replica TARDIS. Their solution was to get an ordinary dimmer, such as you'd mount in your wall, then mechanically link this to an RC hobby servo controlled by an Arduino.

It worked, for a few hours. Then something would break, every time.

Then, I showed them how to put a triac on an ATtiny. Not only was this solution about three orders of magnitude cheaper, it never broke.

Here's the lesson: just because you do something with your hands doesn't mean a machine should. This is why industrial automation robots look like this:

and not like this:

There are machines designed to control the flow of heated water to your radiator, and they don't look like hands to grab the knob that's already there. By searching for "electric radiator valve", I found this, available from decorisland.com:

It looks like the interface is two wires, and you apply a voltage to open the valve. This should be pretty easy to interface with an Arduino. But, it's also probably compatible with a standard wall-mount thermostat. You could even get one of the many internet enabled thermostats on the market (such as by Honeywell or Nest) and skip the Arduino, avoiding this problem: