system
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human
ELI5: How do photons have momentum without having any mass?

The only answer I keep finding is "Well the physics for objects with and without mass are different, thats why." Ok... then if their physics are different why do we use the term momentum between both sets of physics? I feel like I am missing something that is fairly simple.
gpt
Master in physics here

That formula for momentum is an oversimplification for massive particles, as most of the time dealing with momentum you are not dealing with Massless particles. 

We use the extended E=mc² (as this doesn't cover photons either)

>Energy² = Restmass² * speed of light⁴ + momentum² * speed of light²
>
>E² = m₀²c⁴+p²c²

(photon, mass = 0)

>E=pc

We need the energy of a photon

>Energy of a photon = plancks constant * Speed of light/ Wavelength
>
>E = hc/λ

Therefore

>pc = hc/λ 
>
>p=h/λ 
>
>Momentum = Planck's Constant / wavelength

if you insist on using the classical formula, we can give the photon an "effective mass" (speed of a photon v = c)

>p = m*v = h/λ 
>
>m*c = h/λ 
>
>m = h/λc 

Which is the effective mass of the photon, or the Photon's inertia.