1. Fast charging capacitors-Possible? Is the charging speed of a

capacitor determined by the input voltage and current?

The relationship between currnt and voltage for any device that acts

like a capacitor is I=C*(dV/dt) or current equals capacitance times

the rate of change of voltage. The larger the current, the faster the

voltage changes. The large the capacitance, the slower the voltage

changes.

2. I don't understand the differences between these myriad of capacitor

chemistries. But I'm looking at aerogel caps. Does output voltage stay

constant while current varies?

No. The voltage changes at a rate proportion to the current passing

through the capacitor. The voltage may also change instantaneously in

proportion to the magnitude of the current, because the capacitor will

also include some internal resistance.

3. Also, do capacitors accept charge from an a/c source or will it need

a diode?

If you want the capacitor to charge up in one direction, you will need

a diode or other means to keep it from swinging its voltage back and

forth in response to an AC current.

4. LEDs require a resistor, but how do you calculate the required

resistance if you connect the LEDs in series?

LEDs drop a roughly constant voltage over the normal operating range

of current. You select the current you want, subtract the normal

operating voltage from the available voltage, and divide that excess

voltage by the desired current. This will give you a value of

resistance that will consume all the extra voltage when the desired

current is passing through both the LED and the resistor.

4.5. Then the voltage has dropped across each LED, yes?

I think you are confused about the meanings of voltage and current.

Voltage is applied across a pair of nodes, and is a force that moves

charge through the paths between those two nodes. Current is the

moving charge.

5. What determines the capacity of a resistors resistance, i.e., what

is the max voltage/amperage input of a resistor with impedance x. Or do

I have it all wrong and is impedance rated at the specified input

power?

Resistors have a power limit, based on how hot a given power

dissipation causes them to get and the thermal limit of the material

they are made of. They also have a voltage limit based on the

resistor element arching over or other nonlinear things happening.

6. Is the power sent through a resistor merely attenuated or burnt off

as heat?

Current is sent through, power is dissipated within. It is released

as heat into the surroundings.

7. Will a resistor pass proportionally more power through if less than

the power the resistor is rated at is experienced?

The essence of resistance is that it passes a current proportional to

the voltage applied across it. Ohms are a unit that just means volts

per ampere. As long as you keep the product of volts and amperes

lower than the rated power (and briefly if you exceed this) you can

expect this proportionality to be quite constant, but not perfectly so.

8. Can caps handle overvoltage/overcharge (unlike a battery)?

Caps can be destroyed by over voltage, caused by internal arching when

the dielectric layer gets a hole blown in it.

9. Are cap chemistries what determine a caps discharge rate? Are carbon

aerogel caps "fast?" What do they use for railguns?

Not really. The resistance of the electrode system that coats the

dielectric layer determines how high its resistance is, so how much

heat is produced by charging or discharging current, and how much

voltage is used up to create that heat.

And because I'm too lazy to look it up, is there a simple way to figure

how many watts/hour a cap of capacitance xF will put out before

draining?

That's easy. The answer is zero. ;-)

At least if you define draining as any drop in voltage. The capacitor

voltage must change in order for current to pass through it.

The formula for energy in a capacitor is

watt seconds = 1/2 * C * V^2

With the C in farads and V in volts.

Pick a capacitance and a starting voltage and find the starting energy.

Pick and ending voltage and find the energy still left in the

capacitor when the voltage is unusable. The difference between those

two energies is the usable energy you can get from that capacitance as

the voltage falls over that amount.

Be prepared for disappointment.

You will soon find out why chemical batteries have not been replaced

by capacitors except for a few very short duration or very low power

applications.