Fluids #1

An open container filled with ideal fluid starts draining through a spigot, as shown in the image.

Which of these graphs best depicts how height H changes with the fluid velocity v at point 2?

ANSWER:  B.

SHORT-AND-SWEET:

This question requires as little as 20 seconds to answer.  Let's see how.  

What happens as the fluid starts draining?  The height H starts dropping.  What happens to the velocity v of fluid at point 2 as the fluid is draining?  It will decrease as well.  Therefore, H and v track together -- as one goes down, so does the other.  Look at the answer choices.  Answer choices A and C suggest the opposite, and answer choice D suggests that H is constant.  Only answer B suggests the correct relationship between H and v.  Because of conversion of potential energy (static conditions before the spigot opens) to kinetic energy (dynamic conditions once the spigot opens) the actual relationship will be: H = v^2 / 2g.

THE WHOLE STORY:

We start off with an ideal fluid inside an open container.  When the spigot opens the fluid begins draining at a certain velocity, and we can all agree that with that the height (H) of the fluid column starts decreasing.

The question asks about what happens to the height (H) of the fluid as the velocity of the fluid at the spigot (v) changes.

STOP!!!  What I will say next is one of the keys to success on the MCAT, and applies without exception, to every single MCAT section. 

After reading the question, take ten seconds to think about the problem intuitively.  

Chances are that by thinking about the question first, you will be able to guess what the answer should look like.  Then quickly scan through the answer choices, and eliminate those that make no sense.  

The MCAT (and medicine) is as much about picking the right answers, as it is about eliminating the wrong ones.  The reward for this strategy is more precious time left over for questions that actually require calculations.  In fact, most people who end up running out of time in the Physical Sciences section do so because they spend too much time "plugging and chugging" through questions unnecessarily. 

Let's go back to our question and see how we can apply this strategy!

We know that as the fluid is draining the fluid height H will be decreasing.  Initially the fluid velocity v will have some value.  But what happens to the fluid velocity as the container is actively emptying?  With less remaining fluid in the container, the fluid will be draining more slowly (decreased velocity).  

If this doesn't seem right, get a balloon, fill it with water, and punch a whole in the bottom.  Just don't tell your roommates that we told you to do this.

Back to our question.  We have concluded that as the fluid height decreases, the fluid velocity will decrease as well. 

By knowing this, and by looking at the answer choices, we can try eliminating some of the choices.  Choices A. and C. suggest an inverse relationship between fluid height and fluid velocity, which, as we established, is not the case.  Choice D. suggests that fluid height does not change with changes in velocity, which we also know is not the case.  

Therefore, the correct answer is B. 

..........Now, if the answer choices were such that we could not easily eliminate all of the wrong answers, here is the mathematical way to solve this problem. 

What you need to realize is that this is essentially a conservation of energy question.  How?  Before the spigot opens, the system (container with the fluid) is static, because there is no fluid motion.  Once the spigot opens, the fluid will start moving, transforming this into a dynamic system.  

STOP!!!  You should learn that whenever you encounter a question in which you have the same system initially at rest and then in motion, it is very likely that the underlying concept is conservation of energy, focusing on the exchange of potential energy and kinetic energy.  

Let's see how this principle applies here.  For any given fluid molecule its total energy will be the sum of its potential and kinetic energy.  If we take a molecule at the fluid surface (point 1, compared to point 2), given that the molecule is effectively not in motion, its total energy will be its potential energy:  Etotal = Epotential = mgH.

Once the fluid starts draining, by the principle of conservation of energy, each molecule's potential energy will transform into kinetic energy.  At point 2, the molecule will have only kinetic energy, so its total energy will be equal to its kinetic energy:  Etotal = Ekinetic = 1/2 mv^2.

Because total energy is conserved, potential energy at point 1 and kinetic energy at point 2 will be equal:  mgH = 1/2 mv^2

If we divide the whole equation by mass m, we are left with an equation which tells us about the relationship between the height H of the fluid column and the fluid velocity at the spigot v:  

H = v^2 / 2g

This means that the height H of the fluid column is proportional to the square of v.  

You should be able to recognize that what we have in this question is a (parabolic) square function, which is depicted in answer B.

BIG PICTURE:  

1.  Think first, calculate second!  Or, more accurately, think (and eliminate) first, calculate (if needed) second.

2.  Recognize when a question asks you to recognize static ("resting") and dynamic ("moving") conditions within a system.  When they are both in the question, think POTENTIAL and KINETIC ENERGY, and CONSERVATION OF ENERGY!

3.  Know how to interpret graphs, and we mean KNOW IT!