Help pls Further Maths Mechanics Impulse

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keatondocherty
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Can anyone help with this concept?? Take two particles connected by an inextensible string. Initially, the two particles are at rest and the string is slack. One particle moves away from the other at some initial velocity and now the particles are travelling at a combined constant speed with the string now taut.

What I am struggling to understand is why both particles have the SAME velocity after the string goes taut? Surely if both particles experience the same impulse the smaller mass would be travelling at a greater velocity than the other?
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mqb2766
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(Original post by keatondocherty)
Can anyone help with this concept?? Take two particles connected by an inextensible string. Initially, the two particles are at rest and the string is slack. One particle moves away from the other at some initial velocity and now the particles are travelling at a combined constant speed with the string now taut.

What I am struggling to understand is why both particles have the SAME velocity after the string goes taut? Surely if both particles experience the same impulse the smaller mass would be travelling at a greater velocity than the other?
If the string is taut, they must be travelling at the same velocity. If the velocities were different the string would either be slack or broken, as their distance apart would vary.

They experience equal but opposite impulses. The moving mass slows down to the constant speed (negative impulse or acceleratiin). The stationary mass speeds up (positive impulse or acceleration) to the constant speed. The total momentum of the system will remain constant.
Last edited by mqb2766; 1 month ago
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keatondocherty
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Thank you for your reply I understand a little more now. I guess what my question is is do any two particles connected in this way where there are no external forces acting on the system, always reach a common speed? Is it ever possible that..... wait never mind it all just clicked midway through typing this hahaha. As soon as the two particles reach a common speed the tensions in the string would zero and so neither particle would be accelerating anymore. Cheers for your help
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mqb2766
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(Original post by keatondocherty)
Thank you for your reply I understand a little more now. I guess what my question is is do any two particles connected in this way where there are no external forces acting on the system, always reach a common speed? Is it ever possible that..... wait never mind it all just clicked midway through typing this hahaha. As soon as the two particles reach a common speed the tensions in the string would zero and so neither particle would be accelerating anymore. Cheers for your help
Yes. The string becoming taut and the tension being non-zero are instantaneous (impulse) events, like a collision. From a momentum perspective, we're only interested in the steady state behaviour before and after, when there are no forces being exerted on the masses. The (internal) impulse force is instantaneous which "jumps" the velocities.

However, thinking about the impulse from a forces perspective is good for understanding. Newton 1 (inertia - constant speed) applies before and after, as no forces are acting on the masses and they're moving with constant velocity. During the impulse, the two forces are equal and opposite (Newton 3). Newton 2 then says the acceleration is inversely proportional to mass (equal magnitude forces), so that for a light mass the velocity will jump more (to the new, common constant value) than a heavy mass. So momentum scenarios like this are really just Newton 1-3 for "short" duration forces.
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keatondocherty
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(Original post by mqb2766)
Yes. The string becoming taut and the tension being non-zero are instantaneous (impulse) events, like a collision. From a momentum perspective, we're only interested in the steady state behaviour before and after, when there are no forces being exerted on the masses. The impulse force is instantaneous which "jumps" the velocities.

However, thinking about the impulse from a forces perspective is good for understanding. Newton 1 (inertia - constant speed) applies before and after, as no forces are acting on the masses and they're moving with constant velocity. During the impulse, the two forces are equal and opposite (Newton 3). Newton 2 then says the acceleration is inversely proportional to mass (equal magnitude forces), so that for a light particle the velocity will jump more (to the new, common constant value) than a heavy particle. So momentum scenarios like this are really just Newton 1-3 for "short" duration forces.
Yes hahaha thank you. You have no idea how long I was trying to figure this out all yesterday hahaha it was so annoying. I think in further maths they make the assumption that the velocity instantaneously jumps to the common speed because the time the forces are in action is negligible and they also assume that strings in real life are actually inextensible.
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(Original post by keatondocherty)
Yes hahaha thank you. You have no idea how long I was trying to figure this out all yesterday hahaha it was so annoying. I think in further maths they make the assumption that the velocity instantaneously jumps to the common speed because the time the forces are in action is negligible and they also assume that strings in real life are actually inextensible.
Those are "valid" assumptions (of course the modelling of a "collision" is very complex and the simple average force isn't necessarily that accurate), though understanding the large, instantaneous forces (impulses) from a Newton 1-3 viewpoint, means you understand that momentum isn't really something new.

You're simply treating the masses as a single system, and the internal collision forces are equal and opposite, which jumps the velocities as the accelerations (forces) are instantaneous (short duration).

The force determines acceleration. The impulse (force integral) determines the velocity (acceleration integral).
Last edited by mqb2766; 1 month ago
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