Yes but also no
A typical AA battery has a capacity of 2000 mAh. A car battery has ~100000 mAh (100 Ah).
The AA-battery at 1.5 V has about 3 Watt hours. The car battery at 12 V has 1.2 kilo-watt hours.
That's 600 watt hours per shot. Let's be generous and assume that energy is dumped into the target in one second. That equates to 720000 Joule or 720 KJ. The kinetic energy of a typical car moving at 60 mph/100kmh is about 500 KJ.
Solarius, what happens when the human body is hit by a car moving at 100 km/h? Do you think that whatever human-sized target hit by that weapon is going to answer any questions in the forseeable future?
I assume I haven't calculated the calculations, but in fact the entire concept is rather sci-fi, so I assume that the method to project the discharge over such long distances somehow requires large amounts of energy. No, I don't know why this doesn't work like a normal lightning, which is why we have a goddamn not-tony-Stark to figure it out for us.
How about using an elerium battery ?
Because, being able to make a plasma canon and not a light taser canon look really lame.
Yes, an E-115 battery could be much lighter and smaller, but produce the same effect. Good point.
(Not sure how balanced they would be against other ranged stun weapons, though...)
There is another issue with car batteries (and any other chemical batteries). You can't just dump all that energy at once, even if you short circuit them. The chemical reaction needs time to happen. You would actually need capacitors for that. And the problem with capacitors is, that they can't store a lot of energy.
Yeah, that's true as well. Again, I'll dump it on "no-Stark makes it work with high tech". I don't want anything actually silly, but I don't think we should have the exact explanation of every technical detail, as long as it doesn't seem illogical.
Perhaps the 'clip' is actually a smaller chemical battery, custom made by X-Com, and a massive bank of capacitors.
The battery could store more energy than you need to do two shots, but the two shots are all that fits into the capacitors, and they would need too long to recharge. The battery would then be there just to keep them topped off.
That is a plausible explanation, yes.