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When a person walks, he exerts a force on the road, lets say $F$, and the road exerts a force on the legs (friction), lets say $f$. Newton's third law says that $F = f$, so why is $f - F = ma$? Where does that extra $ma$ come from if friction applied is only $F$?

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You are writing that the force on the road minus the force on the leg is equal to mass times acceleration. This is common mistake and is not how Newton's law works. You can se this if you ask yourself what mass and whose acceleartion would you consider in the right hand side of the equation. The leg's or the road? The corrct way to use Newton's second law is to focus on one object at a time. If you consider the leg, then only the forces on the leg should be considered and then you have the net force on the leg is m*a where both m and a reffer to the leg. If you refer to the road, then only the forces acting on the road should be considered. So, to get back to your situation, If F and f were the only forces (in reality there are more forces acting on each object, leg and road) the correct way to write N's second law is f=ma where m is the mass of the "leg" and a is acceleration of the "leg". So there is no mistery about where the acceleration comes from.

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There's some misunderstanding of the law of reaction here. While it's true that $F=f$, those two forces are applied on different entities. A person walking is only affected by the force the road applies to him ($f$ in this case), and not by the force he is impressing to the road.

Thus we have two equations of motion, one for the person, the other for the road:

  1. For the person we have $f=m_{person} \cdot a_{person}$ and hence the person is able to move
  2. For the road we have $f=m_{road} \cdot a_{road}$; now you can not really move the road since it's attached to the soil, so you should be moving the whole tectonic plate, which has an enormous mass, and thus the result acceleration is (basically) $0$.
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Generally, the equation of motion for a body is $$m\vec a = \sum_i \vec F_i$$ When you apply that to walking, nothing special happens. For example, you could write (among many other possible, more or less simplified models of the process) $$m\vec a = \vec F_{muscle}+\vec F_{friction}+\vec F_{gravity}+\vec F_{floor,support}$$ where all the forces on the right hand side may also carry implicit minus signs. Moreover, each force might depend in a specific way on state variables. For example the muscle force depends on the joint angles and what your brain considers appropriate leg-eye coordination. Or the floor support force depends discontinuously on the distance of your foot to the floor ("on-off-force" or more officially, contact force). Same for static friction: the force is restricted to a maximum value, the transgression of which causes a relative velocity (i.e. connected to state variables) between your foot and ground, until only sliding friction remains.

So the $m\vec a$ is not "extra", it is just the core of Newtonian mechanics.

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While others have explained the misconception you have about Newton's 2nd and third laws, they still haven't given a hint about the mechanism of walking which is in your title.

In order to walk, you must begin to fall forward. That means you begin to rotate around some point which is your (hopefully) stationary foot. As your center of mass moves in front of that foot, because of the internal actions of your muscles and bones, gravity exerts a torque and you begin to rotate. You move your non-stationary foot in front of your center of mass and plant it firmly. You change the net force on the ground from your previously stationary foot so that the net ground force exerts a torque about your new stationary foot-ground contact point to keep your body rotating in a forward motion. This gets repeated and you walk. It's a repetitive falling/stopping/falling motion. Some people learn to do this well, while others plod along clumsily, but the motion is caused by alternating gravitational and ground-based forces producing torques about your stationary feet.

If your feet slip, bad things can happen!

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As our muscles allow our center of mass to move forward past the points of contact between our foot and the ground the friction keeps our foot from sliding out from under us as the muscles push it further behind. Then we move the other foot further in front of us to repeat the process as our momentum and muscles carry our center of mass onward. Were we standing on ice with a much lower coefficient of friction our foot might overcome static friction and slide out from under us when our muscles attempted to shift our center of mass forwards. Hope this helps.

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