crankshaft wave 125

If a welder can briefly output more power (i.e. 100A x 50V= 5kW) than its rated input (13A x 115V=1.495kW), does it use a battery or capacitor to store up reserve power?

You said correctly: VERY BRIEFLY…. as soon as the electrode arc allows the welder to deliver its full current capacity, its impedance makes it drop the voltage to lower values... and there goes those imaginary 50 volts after a very brief time! So forget about maintaining those "50 Volts"... Those will kwickly drop to a much lower value, thus never reaching that imaginary 5 Kilowatt rating anymore. And at the high currents that they operate, their rating goes down, so that they need to specify a DUTY CYCLE… that means that for every minute of welding, the welder has to cool down for a period of time, because it will overheat (in the case of transformer types). Newer “Inverter” power supplies can perform better but are more expensive and some can be very delicate. "Rated Power" needs to be carefully understood... There have been numerous cases of badly overstating such "ratings". Back in the golden days of High Fidelity Power Amplifiers, there were different ways to rate the true power output, as the "Watts per channel" race was in its most intense heat, and price was mostly based on those "Watts per Channel" basis. At least many honest manufacturers gave the measurable and repeatable values of "Sine-Wave continuous into 8 Ohm" or "RMS Watts" ratings that while not really correlated to music reproduction demands, at least were easily measurable and comparable among brands and models... But soon, even suppossedly reputable brands as SONY or PANASONIC (and many more manufacturers) that also produced Mid an LOW Fidelity music centers, started to blatantly LIE to consumers, stating that their absolutely trashy products were capable of absurd ratings of THOUSANDS of Watts per channel! Those useless values are as inflated and misleading as saying that and old unmoddified 4 cyl opposed engine VW Bug, can truly produce 1,000 HP... IT can indeed, but to make such power you had to press the clutch pedal, accelerate the poor little engine to redline RPMs, and just before exploding in pieces, release the clutch suddenly... the poor little car would spin the tires for a brief fraction of a second, and the engine would die and stop. But for that very brief fraction of a second, you could claim it just produced a thousand Horsepower! Completely useless and misleading. Well, the ratings on some welders are like that. At least the VW Bug engine can store some the energy because of the inertia of the crankshaft and rotary parts, like the small flywheel and that's all, BUT welders do not store energy to validly claim usable larger than real power. Even the rated input power is subject to the Thermal Limits of its components, so it is important to take that into account. A homeowner or hobbyist welder that works from a single phase 110 to 125 volt AC receptacle will be usually limited to brief cycles of full power, and will overheat soon. Good for small home Jobs where you can wait, take a break, go for a drink, allow the welder to cool down, and then return to work a little more, but unusable in a small trade shop!