*Ryen asked the Naked Scientists:*

We were debating at work if over time the weight of the Earth increases due to photon bombardment from the Sun. Then we found that apparently photons have no mass, yet they have momentum. Could you please explain how this is possible?

*What do you think?*

At school you learn that momentum p is equal to mass multiplied velocity, but that formula

is not correct, it's only

*approximately* correct, for objects with mass, at low speeds, and it's wrong for objects without mass.

The correct equation is:

p = Sqrt[(E/c)

^{2} - (mc)

^{2}]

E = total energy; m = mass

In that equation, valid at any speed and for massive or massless objects, put m = 0 (the case of photons) and you will find p = E/c; so photons do have momentum even if they are massless.

From an empirical point of view, switch on a very powerful laser in a certain direction, and you'll be able to measure a recoil in the opposite direction; this is the experimental prove that light has momentum.

If you want to find the approximated formula p = mv at low speeds for massive objects, write the above equation in this way (after some computations):

p = Sqrt[2mE

_{k}(1 + E

_{k}/2mc

^{2})]

where E

_{k} = kinetic energy = E - mc

^{2}At low speeds, E

_{k} << mc

^{2} so you can neglect E

_{k}/2mc

^{2} obtaining:

p ≈ Sqrt(2mE

_{k}) ≈ Sqrt(2m*mv

^{2}/2) = Sqrt(m

^{2}v

^{2}) = mv.