Does it matter if power is applied only instantaneously?
One million watts for 1 picosecond is only 1 micro Joule, not a lot of energy. If the conversion of electrical  energy to kinetic is 100% efficient, then you can work out through \$\frac{1}{2}mv^2\$ how fast your projectile is going to be moving after acquiring 1&nbsp;&micro;J.

However, we don't tend to power railguns with a specific power for a specific time. Where would you get a one megawatt power supply? Instead, we tend to build a storage system, maybe a bunch of capacitors, maybe a spinning alternator-based machine called a compulsator, that stores a certain amount of energy. Then the challenge is to dump that energy store into the railgun load in the right sort of length of time, which generally involves controlling the inductance of the supply, switches, cables, everything really. So you'll start with an energy, not a power, and work from there.

Let's do some sums with some real data. Let's say you put together a number of capacitors, say from the CD942 range, very high current capability, probably overkill, but useful to get ballpark figures. A 2kV 150nF cap stores 300 mJ when fully charged, and occupies a space of around 30x30x50 mm. So 1000 would fit into a milk-crate sort of volume, and you'd be able to lift it (I wouldn't). That will store 300 J. If discharged in 100 uS, that would deliver a power of 3 MW.

That's still rather on the low side for stored power for a railgun, which scales to small sizes very badly. Have a look at this paper for a railgun supply, where they talk about mega-Joule levels of stored energy, delivered over a few milliseconds.