Physics Olympiad 2010 practice test question # 23

[NewtonGalileo]

Two streams of water flow through two U-shaped tubes (the two U-tubes are facing away from each other, but touching each other at the base of the 'U's). The tube on the left has cross-sectional area A, and the speed of the water flowing through it is v. The tube on the right has cross-sectional area A' = A/2. If the net force on the tube assembly is zero, what must be the speed v' of the water flowing through the tube on the right? Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit.

 

[tiny-tim]

Hi NewtonGalileo!

What principle do you think you should use?

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[NewtonGalileo]

This is what I tried.

Imagine that the two U's were separated and were moving towards each other. Since their final momentum is 0 (they're touching each other in equilibrium as the problem states), their initial momentum ought to be 0 because the system is closed. So, since the tube on the left has twice the mass of the tube on the right, the speed of water on the right has to be twice the speed of water on the left. So, my answer was $$2v$$. But, this solution really does not make much sense to me, and the correct answer should be $$\sqrt{2}*v$$. Please help!

 

[tiny-tim]

(have a square-root: √ )

how did you calculate the momentum?

 

[alemsalem]

The momentum of each tube is zero, they don't have equal and opposite momentum which cancels (they do but it's zero because no tube is moving),, so for each tube water is coming in and leaving at the same speed, for this to happen there would have to be a force on the tube which is canceled from the force due to the other tube..

 

[tiny-tim]

hi alemsalem!

The momentum of each tube is zero …

uhh?

the momentum changes: it start to the left, but after the bend it's to the right …

so how much does it change?​

 

[alemsalem]

hi alemsalem!


uhh?

the momentum changes: it start to the left, but after the bend it's to the right …

so how much does it change?​

I was referring earlier to his approach, he treated it as if the two U tubes where moving in opposite directions and one with twice the mass should have half the velocity in order to have equal and opposite momentum. but that's not the case the total momentum of each tube alone is zero

 

[tiny-tim]

hi alemsalem!

I was referring earlier to his approach, he treated it as if the two U tubes where moving in opposite directions and one with twice the mass should have half the velocity in order to have equal and opposite momentum. but that's not the case the total momentum of each tube alone is zero

not really following you (or maybe I'm not following him)

anyway, your calculations are correct, but please don't give the full answer when the OP hasn't yet done the question …

fortunately, he's still off-line, so can you do a bit of editing?

 

[alemsalem]

done