# Relationship between speed and flow

#### Uricucu

Hi! I'm currently doing a project to simulate a water slope, and want to find the maximum speed of a person going downhill in function of the water flow. At first I thought using a combination of the Manning equation and the Newton Law for viscosity, through the sheet of water, but I'm in a dead point...

Does anyone know any group of formulas to establish such relationship? Thanks!

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#### anorlunda

Mentor
Gold Member
Welcome to PF.

I'm not sure I understand your question. People on water slides can travel faster than the water flow. They even have slopes where the water moves up the slope (negative water speed), while the people on surfboards go down slope.

#### Uricucu

Welcome to PF.

I'm not sure I understand your question. People on water slides can travel faster than the water flow. They even have slopes where the water moves up the slope (negative water speed), while the people on surfboards go down slope.

Thank you for responding. Actually I'm trying to calculate the speed of a person in a water slide, in particular the descending speed in a slope. I guess the person speed is related to the water flow and the degree of the slope, but I was asking for any formula that relate both.

Hope I could clarify things, if not, feel free to ask!

#### anorlunda

Mentor
Gold Member
related to the water flow
Wouldn't it depend on depth of the water also? I think of a slide with a wetted surface <1 mm thick. The film of water hardly moves at all but the person sliding can go very fast.

But if the water is deep enough, then the person is "surfing" with no contact between his body and the slide itself. Similarly, the person's body position would also be a factor.

I guess what I'm saying is that I doubt that there is one simple formula. But I have been wrong many times before, so perhaps other PF members can prove me wrong.

#### Uricucu

Yep, I thought about the Newton viscosity law, that links the velocity with the water thickness. But that thickness is the one beneath the person, so its like 0,01 mm (logically). My idea is to know the flow in order to not exceed the maximum velocity, which is 14 m/s.

#### russ_watters

Mentor
Wouldn't it depend on depth of the water also? I think of a slide with a wetted surface <1 mm thick. The film of water hardly moves at all but the person sliding can go very fast.
It might, but I was thinking at least for a thin film moving with the person, the coefficient of friction is so low you can ignore it and just use gravitational acceleration.

#### gmax137

the coefficient of friction is so low you can ignore it and just use gravitational acceleration
That's what I was thinking. Then, to limit the top speed, you have to limit the height (or limit the height in any one continuous downslope). For 14 m/sec I get 10 meters max change in elevation.

I never thought of it before, but I guess that's why the slides seen in waterparks always seem to have "flat spots" -- to limit the top speeds.

#### Uricucu

So the speed is more related to the slope than the water flow? But it's still related to water flow in some way, isn't it?

#### Uricucu

Hi again, thank you all of you for responding my post. I don't know if I get the main ideam but what you all seem to agree is that I should apply the basics of physics, meaning the formulas like: x=xo+vo*t+0.5*g*t^2 and v=vo+gt?

Then, the mass of people and their area aren't a variable that effect their velocity, are they? Moreover, do you think that the water speed has any effect at all in the user's speed?

Thanks.

#### anorlunda

Mentor
Gold Member
Then, the mass of people and their area aren't a variable that effect their velocity, are they? Moreover, do you think that the water speed has any effect at all in the user's speed?
Search for Galileo's experiment. He dropped a heavy ball and a light one at the same time. They landed at the same time. So no, mass does not change the speed of falling.

Area is different. A feather falls more slowly because of air drag.

"any effect at all" is not the right way to engineer this. Engineers deal with "any effect big enough to worry about?" and to that, the consensus is no. Water speed is probably not big enough to worry about, and air drag at speeds less than say 50 mph are also too small to worry about.

Also, a good engineer always remembers about variations, and factors not considered that limit the accuracy of calculations. You should aim for an estimate of speed to the nearest 10 mph, not 0.01 mph. Then add a range to that, such as 40 mph $\pm{10} mph$.

#### Uricucu

So maybe I've been focusing in the relationship between the water flow and the water depth with the user's speed, while I should have been focusing in the air effects, since you adivsed me to neglect water friction effects in terms of calculations.

Guess I will try to limit the terminal velocity to the effect of the air drag.

To context you all, since I think I haven't been clear at all, my college project consists in designing a water slide, from the design itself to the steel structure. Now I'm focusing in calculating the user's speed in order to guarantee they don't exceed the maximum allowed, which is 14 m/s.

Thanks!

#### JBA

If you are going to start by ignoring the water effect then a very basic start would be to calculate the maximum speed based upon the acceleration of gravity for the planned height and slope of your slide and then compare that to your terminal velocity calculation. At the same time, since water inserted at the top of the slide will achieve the same velocity, except for the slide surface drag effects, you might use the standard equations for sloped channel flow to estimate the water's velocity to see if there is a significant enough differential to your first calculation to see if an investigation of the water drag on your rider would be worthwhile.

"Relationship between speed and flow"

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