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As I remember, on old day explanation, a person in a car hitting the interior of the car cannot move the car, because there is no action and reaction force between the car and ground. But I doubt the car can move actually because:

  1. When the person starts to extend the arm, due to conservation of momentum between arm and car, the car moves to the opposite direction of arms

  2. When the arm actually hits the interior, the arm stops, and due to conservation of momentum, the car stops.

  3. In the time between the person starts extending the arm until the arm hits the car, the car has displacement

So I think if the person hits the car and don't retract the arm, the car would have net movement, am I correct?

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The car can be moved provided it isn't on a frictionless surface, or more precisely the centre of mass of the car/occupant system can be moved provided the car isn't on a frictionless surface.

As you say in the question, assuming no external forces are acting then because the momentum of the car + occupant is conserved the occupant cannot move the centre of mass of the car + occupant.

However if this is a real car on a real road then we need to include the car/transmission/road friction. This means that a gentle movement will not move the car because the force produced is smaller than the friction. However a violent movement can produce a force bigger than the friction. So if the occupant moves gently in one direction and violently in the other direction this can produce a net motion.

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  • $\begingroup$ "The car can be moved provided it isn't on a frictionless surface." As the question is phrased with the arm being extended and not retracted the car will move backwards particularly if there is no friction because the centre of mass of the arm has moved forwards and the car and occupant system centre of mass has not moved. $\endgroup$ – Farcher Jul 27 '16 at 8:53
  • $\begingroup$ Sorry I pressed the return button too early. $\endgroup$ – Farcher Jul 27 '16 at 8:56
  • $\begingroup$ @Farcher: OK I've clarified the opening statement to make it clear what I'm saying. $\endgroup$ – John Rennie Jul 27 '16 at 9:01
  • $\begingroup$ I am tempted to test this by scooting my chair across the office without touching the floor ... $\endgroup$ – RedGrittyBrick Jul 27 '16 at 9:04
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    $\begingroup$ Using a variation of the methods employed by Farcher et al, I have experimentally verified the proposition of Dr Rennie! None of my colleagues actually noticed - but they are not physicists. $\endgroup$ – RedGrittyBrick Jul 27 '16 at 9:17
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This is a two-body problem with the center of mass at rest, much like a stationary nucleus emitting an alpha particle. Therefore if the person were to move in one direction the car would go in the opposite direction.

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seeing the person and the car as one system you can say that the center of mass always stays at the same point if no external force is applied.

However, if the person jumps in one direction inside the car, the car will move in the other direction. The center of mass stays at the same position. Of course the floor applies an external force to the system and therefore manipulates the system. In space it would work in any direction.

Just see fireworks. Fireworks fly on a curve. After the explosion the center of mass of all the small particles still follows the exact same curve.

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No the car cannot move... even if it is in the friction less surface. and the force you give to the car with a hit in inside is an internal force if you consider car and road as the system. as the internal force have newton third law counter forces within the system they cancel out and the net force is 0, lets say you are hitting the dashboard with force F here the opposite force is given by dashboard with same magnitude as F so they cancel out and car does not move

if a force is externally given to the system like the newton third law counter force act out of the system (which we dont have to bother)

so the both situation differ the internal force given to the system does not move the center of mass of system but external force given to the the system will move system's cnter of mass

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