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This question already has an answer here:

According to Newton's third law of motion, if body A exerts a force on body B, then body B exerts a force on body A that is equal in size but opposite in direction, then how do things move?

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marked as duplicate by garyp, Wolpertinger, user36790, sammy gerbil, Javier Aug 16 '16 at 14:59

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    $\begingroup$ In your picture, only one force acts on each body, so it moves. $\endgroup$ – Valter Moretti Aug 16 '16 at 11:39
  • $\begingroup$ When the wheels of a car gain traction against the road, a reaction force is exerted on the Earth. But because the Earth is much larger that the car, so the car moves but not the Earth, at least not perceptibly (Newton's second law). $\endgroup$ – Gert Aug 16 '16 at 11:46
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Bodies move because there are two equal and opposite forces on two different bodies, not on the same one.

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    $\begingroup$ But if I push a cupboard the cupboard moves but my hand doesn't move in the opposite direction $\endgroup$ – Ziva Aug 16 '16 at 11:49
  • $\begingroup$ Do you mean to say you feel no force on your hand when you push a cupboard?! You do have the rest of your body attached to your hand, you know--perhaps it has some part to play in your hand not moving! $\endgroup$ – Gauri Aug 16 '16 at 11:51
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    $\begingroup$ @Ziva If you did it while jumping or standing one ice you'd move backwards. If you're standing on normal, you'll instead pass on the force onto the ground. But since the ground is so massive, the force you exert on it only results in a tiny acceleration. $\endgroup$ – CodesInChaos Aug 16 '16 at 14:54
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    $\begingroup$ Try pushing the cupboard while standing on a skateboard. $\endgroup$ – Chris Cudmore Aug 16 '16 at 17:10
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In your picture, only one force acts on each body, so it moves or better it has acceleration according to $\vec{a}= \frac{1}{m}\vec{F}$.

It is true that the sum of all forces is zero. This has a consequence only for the whole system made of the two bodies: Since the center of mass of the system always moves in accordance with the second law of dynamics where $\vec{F}$ is the sum of all forces acting on each part of the system, the center of mass has here vanishing acceleration and thus it does move if its initial velocity is zero.

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