stitchattack

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As water is pumped through a series of pipes, it travels from pipe A of radius r directly to pipe B of radius 1/4 r. Which of the following is true?
A. the amount of volume flowing through pipe A per unit time is 1/16 of that flowing through pipe B
B. the amount of volume flowing through pipe A per unit time is 16 times that flowing through pipe B.
C. the amount of distance the water travels in pipe A per unit time is 1/16 of that in pipe B
D. the amount of distance the water travels in pipe A per unit time is 16 times that in pipe B

I know that by using the continuity equation A1V1 = A2V2 you get:
(pi)r^2 = (pi)(1/4)^2
(pi)r^2 = (pi)(1/16)

I picked A but the answer is C. I'm obviously confusing on this concept. Why is it that the answer is on the "amount of distance the water travels" versus the "amount of volume flowing through"? My take on the continuity equation is that it tells the volume flow rate. Thanks!
 

justadream

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@stitchattack

I am by means no expert on fluids but here's my take:

The continuity equation assumes CONSTANT volume flow rate (AV = CONSTANT). No matter what, the total amount (by that I mean volume) moving through is constant (after all, the pipe is not leaking and there are no tubes feeding into it).

So if the radius of pipe B is smaller, then to keep the "AV" term constant, you need to increase velocity in pipe B.
Because area has the radius squared term, you get that the velocity in pipe B is 16 times faster than the velocity in pipe A.

That's the same thing as saying the velocity in pipe A is 1/16 that of pipe B.

My statement above is equivalent to saying the distance traveled by pipe A is 1/16 that of pipe B since velocity is directly proportional to distance traveled.

For example, say you are going 16mph and I am going 1 mph. That means in any given time period, you will travel 16 times the distance that I do.
 

Chrisz

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The answer is indeed C. A1V1 = A2V2. If A2 is 1/16 of A1, then V2 is 16 times faster than V1. Amount of distance per unit time is simply velocity.
 
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America

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You should think of it as "the total amount of flow or the total volume is constant",
On the MCAT all fluids are ideal unless otherwise indicated.
Since ideal fluids are incompressible, their volume remains constant. ( A & B are out)

The volume of a fluid moving through a section of pipe is given by A(cross-sectional area of the pipe) times the distance (d) of the pipe section.

Since the fluid moves a distance d in time t, its velocity is v=d/t. Putting equations together you get the continuity equation: Q =Av

Don't forget: at the smallest cross-sectional area, the velocity of flow will be greatest and the pressure will be lowest.
 
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