Capacitor sizing for Time off Delay Formula?

Is that 40 sec or 40 msec? 40 seconds will take a very large capacitor unless you are talking a very small motor. Normal "big" capacitors don't store much energy at all.

Energy in a capacitor is 1/2 C V^2

If you want a 1 watt motor to turn for 40 sec, that's 40 watt seconds of energy or 40 joules.

At 12 volts that's:

40 = 1/2 C 12^2

so C is .55 Farad

Or 550,000 uF - a very large capacitor.

I'm not completely sure I got all the units above correct BTW.

And this only gives you the lower limit of capacitor size because voltage drops as the charge drains so the motor will slow down and stop at some point long before all the energy is actually used. So you will need a much larger capacitor (10x the size) if you actually want to keep driving the motor near the same speed without the help of a regulator.

And of course, the wattage rating of the motor is not the same thing as the actual watts being used for a given application. You need to measure the current and voltage to calculate wattage while the motor is running in your application under the actual load you want to keep driving for 40 seconds.

I realized there's a fairly easy way to do the calculation to deal with the fact the voltage is dropping while the charge drains from the capacitor. You just need to to first determine how much voltage drop is acceptable in your 40 second time frame. If the answer is that the voltage can drop from

12, to 10 volts, but no further, then the capacitor will need 10 volts of charge after 40 seconds. You can calculate how much energy is left, and how much was needed at the start, and directly determine the capacitor size that way.

So, again with the same numbers as the above example:

40 j = start - end 40 = 1/2 C 12^2 - 1/2 C 10^2 40*2/(12^2-10^2) = C = 1.8 Farads, or 1,800,000 uF

Are you really trying to put enough of a charge on a capacitor to keep the motors turning for 40 seconds? I don't know what your actual project is, but this seems very unlikely as the best way to accomplish it. Why not put a 40 second timer on a relay?