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Suppose there is a rope which is initially untaut, two people go and hold both ends of the rope and start running in opposite directions at constant velocity. Let person holding one end be '$A$' and the person holding other be '$B$', now when the rope becomes taut and the people try to move further, would there be a tension force exerted between the two people $?$

I am confused if it there would be because by definition they are moving at constant velocity but this mathematical picture defies my intuition that they should exert some force on each other

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They simply cannot move at constant velocity relative to one another once the rope becomes taut, because past that point the rope would have tension that exerts a force on each runner, causing each of them to (rapidly) decelerate, and probably fall down. If they do not fall down, then just after the rope gets taut, the runners still continue moving a little bit apart due to momentum, hence stretching the rope a bit, and then it will reach some point where the runners have zero velocity (relative to each other), at which point the rope will be slightly longer than just taut, and will pull them towards one another.

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  • $\begingroup$ Actually they can move with constant velocity, if magnitude of linear momentum of any one person say A is greater than the other, then after the string becomes taut the whole system( person A + person B), will move towards the direction A is moving, with a constant velocity , as the string becomes taut $\endgroup$ – maverick Jul 27 '20 at 8:19
  • $\begingroup$ @maverick: You are right. I correctly said "zero velocity relative to each other" later in my post, but forgot to say that at the start... Let me fix it and thanks for pointing out! $\endgroup$ – user21820 Jul 27 '20 at 8:26
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the force here is due to momentum change

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would there be a tension force exerted between the two people

Yes , they are trying to move apart but since the rope can't be stretched there is a tension force on the people towards their COM.

As soon as the rope becomes taut there will be a change of momentum as the rope will exert a tension force their respective velocities will change.

https://youtu.be/RBQ8ywSd7R4

Look at the first 5 seconds of this video. The situation is of familiar to what you are asking. The truck tries to move away but the tension stops it.The only difference is that here the tree and tension both will try to stop the truck.

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