How can photons have momentum if they don't have mass?

p=mv so if the photon has 0 mass then shouldn't it have 0 momentum?

This is one that's stumped me for a long time, and now finally I think I've found the answer. It's to do with Einstein's E=mc^2.

Rest masses

This concept was first proposed by Einstein in his Special Relativity. A rest mass is the mass of an object when it isn't moving at all, it has no velocity. Special relativity says that when an object is moving it's mass increases. So technically speaking you're heavier when you're walking around than when you're stood still, but the effect is ever so small. So because the photon is constantly moving at the speed of light, technically it has mass, even though it's rest mass is 0. And a photon never slows down so technically it always has mass. But it also technically doesn't.

What's that got to do with e=mc^2?

Well we can say the photon is also a wave (wave particle duality) and we can use the formula E=hf where E is energy, h is Plank's Constant and f is the frequency of the photon. From this we see that the photon has energy and from E=mc^2 we see that energy can also be mass.

Now it's time to go off on a bit of a tangent

Actually when Einstein published his new theory back in the day, He wrote the formula as m=E/c^2 and not E=mc^2. This is because his theory wasn't trying to say that energy and mass can be converted from one to the other (that's actually kinda not true...), he was trying to say that mass is actually a form of energy. Cool hey? (or not :/ )

I think I got this mostly right... (I hope)

If it's wrong let me know and I'll change it :). Hope you enjoyed reading and maybe learnt something new :0