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Momentum delivered to a piston

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Okay so please forgive the simple question - I am slowly educating myself but find that sometimes the intuitive view evades me.

In Feynmans Chapter 38 of Vol 1 - I am reading through kinetic theory of gases about the momentum of a particle hitting a static piston. The section after 39.2. He states that in the case of a particle with mass and velocity hitting the piston with a an elastic collison the particle is reflected. OKay. And goes on to say the the total momentum delivered to the piston is the momentum 'in' and the momentum 'out'. So I understand that the total momentum in the system is 2mv but I'm having difficulty understanding why the momentum 'delivered' to the piston is not just mv. (Also I am unsure if the momentum 'out' refers to the particle reflected and its momentum or the momentum of the piston after the collison - I assumed the reflected particle because I think the piston is locked in place).
I know I'm just missing something stupid as all sources do the same work up - but I still intuitively can't see why the Force acting on the piston is not just the mv. Please help me reach enlightenment!

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