If I push a block on a frictionless surface with enough force, it begins to move. Shouldn't it be standing still if it exerts an equal force on me as per 3rd law?
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Exactly. The block exerts a force on you, not on itself. You apply a force on the block. The block applies the same force on you. Because you have feet, your muscles act so that you do not move back (of course not on a frictionless surface :P). The block has no such internal forces to counter your push. It can have a little resistance if the surface has friction.