Explaining equation to stem class

[bkelly]

I am a volunteer teaching a STEM class once a week to a class of eight elementary students. From this page http://tbpmindset.org/modules/SECME/Rockets/Rockets_Calculations_Manual.pdf
I use this equation for water flow out of a water rocket
m = nozzle_area * discharge_coeficient * SQRT( 2 * water_density * guage_pressure )
(I used my own words to help the equation stand alone here.)
Question: How should I explain the constant 2 and taking the square root of the last three terms?

 

[atyy]

I think it's is primarily from the kinetic energy term in Bernoulli's principle, which is based on energy conservation. [/PLAIN] [Broken]

http://en.wikipedia.org/wiki/Bernoulli's_principle[/URL] [Broken]

$P_{1} + \frac{1}{2} \rho V_{1}^{2} + \rho g z_{1} = P_{2} + \frac{1}{2} \rho V_{2}^{2} + \rho g z_{2}$

Assume the velocity is the same throughout the pipe: $V_{1} = V_{2}$
Assume the pipe is level: $z_{1} = z_{2}$

$V = \sqrt{2(P_{1} - P{2})/ \rho}$

The velocity $V$ should be related to the rate of flow of mass $\dot{m}$.

 

[Andy Resnick]

I am a volunteer teaching a STEM class once a week to a class of eight elementary students. snip>
Question: How should I explain the constant 2 and taking the square root of the last three terms?

Can you provide some context? For example, what grade is the class- do they know what a square root is? Do they have any understanding of units?

 

[bkelly]

Context! I should have included that in the OP. They do know what square root is, and at least fundamentals of exponents. I am unsure about units, but I do explain that very carefully in the first few sessions on bottle rockets. We spent several of our weeks taking apart gas engines and discussing how they work. The understand the area on the top of the piston and the volume of the cylinder. They are fifth grade and they are far ahead of where I recall being in fifth grade.

I should also say that I can logically figure out the terms of the equation and why they are present but I do not understand the constant 2 and the square root. I am a software engineer but clearly not a mathematician.

I have been reading the linked Wiki page and am not able to understand it. I don't want to go into that detail with the students. I just need a basic concept of the constant and the square root.

 

[brainpushups]

Bernoulli's equation is essentially a statement of conservation of energy. Compare
1/2 m v²+ m g y = constant (a statement that kinetic energy plus gravitational potential energy near Earth's surface is conserved) to
1/2 ρ v² + ρ g y + P = constant
and notice the similarities

The only difference two of the terms is that the density is used instead of mass in Bernoulli's equation.

So, from a standpoint of units, Bernoulli's equation expresses energy per unit volume (If you look at the base units of pressure you'll notice that they are equivalent to units of energy per unit volume). The pressure term P is only necessary when there is a difference in pressure. If you have an open container that term is not necessary (just FYI).

So... your question essentially boils down to 'why is kinetic energy dependent on the square of velocity?' and 'why does kinetic energy have a factor of 1/2 in front of it?' There is not a quick answer for this because there is a lot of history behind it. The gist of it is that Leibniz first noticed that the square of an object's speed was proportional to the height from which it fell; he called this 'force of motion' vis viva and defined it as m v².

The 1/2 came from Coriolis who defined work as Force*distance. The 1/2 is the factor that connects vis viva to work. Later, this 1/2 m v² was called kinetic energy by Lord Kelvin. Perhaps the easiest way to explain the 1/2 is to explain that it would not be 1/2 if you were working in a different system of units. The 1/2 doesn't need to be there, but without it there would be a whole lot of 2s popping up in other formulas (ironically, your problem is that there is a 2 in your formula!). It is a matter of convenience for the overall structure of classical mechanics.

I dunno. I have a hard time explaining this to adults. I don't know how I would approach it with 10 year olds! Good luck!

 

[bkelly]

I kind of follow along, but have a difficult time. I will look at this some more. In the mean time, my best option is to say that people who understand this have figured it out and we are using the fruits of their labor. If you, the children, want to pursue this I will help, but I will also be trying to understand it along with you. Meanwhile, let's calculate what we can and launch a few rockets.
Thank you for your time

 

[atyy]

Are you familiar with the expression that kinetic energy is (1/2) X mass X (velocity)²?

That is where the 2 comes from, and the square root, by inverting the equation to solve for the velocity.

http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html

 

[bkelly]

reply to atyy: No, not comfortable with it. That site does not explain well. It asks question then does not answer. On the second page the author posts the equation d = vt = v_f t / 2 Except that the first v has a bar over it. Ok, what is that v-bar. I don't see the explanation.
I have been visiting other sites and have not yet figured it out.
I will keep looking.

 

[brainpushups]

vbar is the average velocity. The example in the link derives the formula for kinetic energy by assuming the acceleration is constant so the kinematic equations can be invoked, but the result is general. Because (and only because) the acceleration is constant, the average velocity is simply the arithmetic mean of the initial and final velocity.

 

[vela]

Question: How should I explain the constant 2 and taking the square root of the last three terms?

The equation given on the web page is the result of some algebraic simplification, which obscures the meaning of the various factors. It would be a bit clearer if you explained the mass flow rate is given by
$$\dot{m} = cd \times \rho \times A \times \sqrt{2 \Delta P/\rho}.$$ As atyy explained, $\sqrt{2 \Delta P/\rho}$ is the velocity of the fluid, which follows from Bernoulli's principle, which is really conservation of energy. The cross-sectional area times the velocity gives you a volume flow rate, and if you multiply the volume flow rate by the density, you get a mass flow rate. Finally, the factor $cd$ is a fudge factor to account for the shape of the nozzle.

Concentrating specifically on the factor of 2 and the square root seems to me to be misguided. They're there as a consequence of the math.