Two identical bullets are fired one by a light rifle and another by a heavy
rifle with the same force. Which rifle will hurt the shoulder more and why

Dear Student,
As mentioned in the question, if the bullets are fired with the same ‘force’ that means the bullets leave the guns with equal velocity. Let their masses be ‘m’ and their velocities be ‘u’.

Let the mass of the light gun be m₁ and the mass of the heavy gun be m₂. After firing let their velocities be v₁ and v₂ respectively.

Considering momentum conservation for the light gun,

Momentum of the system before firing = Momentum of the system after firing

=> 0 =  mu – m₁v₁

=> m₁v₁ = mu

Considering momentum conservation for the heavy gun,

Momentum of the system before firing = Momentum of the system after firing

=> 0 =  mu – m₂v₂

=> m₂v₂ = mu

Thus, momentum of both the guns are same after firing, hence, both will equally hurt the shoulder.
Regards