At what speed are quadratic and linear drag forces equal for a baseball?

[Oblio]

When a baseball flies through the air, the ratio f$$_{quad}$$ / f $$_{lin}$$ of the quadratic to the linear drag force is given by

$$\frac{f_{quad}}{f_{lin}}$$ = $$\frac{cv^{2}}{bv}$$ = $$\frac{\gamma D}{\beta}$$ v = (1.6 x 10$$^{3}$$ $$\frac{s}{m^{2}}$$) Dv.

Given that a baseball has a diamater of 7 cm, find the approximate speed v at which the two drag forces are equally important. For what approximate range of speeds is it sage to treat the drag foce as purely quadratic? Under normal conditions is it a good approximation to ignore the linear term?



f$$_{lin}$$ = bv

f$$^{quad}$$ = cv$$^{2}$$



Dumb question in starting this, does the b here represent slope? I can't find a definition of the variable in my text...

The Attempt at a Solution

 

[Oblio]

I found that

b = $$\beta$$D

c = $$\gamma$$D$$^{2}$$

I'm not sure how the ratio works so that you find which force is negligable and which is not.
Help?

 

[learningphysics]

For the first part, at what velocity in terms of b and c, is the quadratic component equal to the linear component ?

my guess is, it's safe to ignore the linear term if the quadratic force is ten times the linear force or more.

 

[Oblio]

How does that equation though, since the two thing being compared and being divided, show what can be ignored?

 

[learningphysics]

How does that equation though, since the two thing being compared and being divided, show what can be ignored?

It's showing you the ratio of the two forces... if the ratio is high then the linear term is insignificant compared to the quadratic term (since the quadratic term is so much bigger)... so the linear term won't have much of an effect as compared to the quadratic term... so we might as well ignore it.

On the other hand if the ratio is extremely low... close to 0, then the quadratic term is insignificant compared to the linear term... and we can ignore the quadratic term, and keep the linear term.

 

[learningphysics]

I used a ratio of 10, purely as a guess... I don't know what would be a good ratio... 10 times seems big enough...

 

[Oblio]

I see...
So, i want it equal to 1 for them to be equally important?

 

[learningphysics]

I see...
So, i want it equal to 1 for them to be equally important?

yes, exactly.

 

[Oblio]

Ok but the velocity is always squared in the quadratic force. How can I ever change the velocity to make them equal?

 

[learningphysics]

Ok but the velocity is always squared in the quadratic force. How can I ever change the velocity to make them equal?

Solve for the velocity. You'll see.

 

[Oblio]

I got 9.14 x 10^-3 m/s... that doesn't sound right.

 

[Oblio]

the values given for beta is 1.6 x 10^-4 Ns/m^2

and gamma is 0.25 Ns^2/m^4

 

[learningphysics]

I got 9.14 x 10^-3 m/s... that doesn't sound right.

What is b and c?

 

[learningphysics]

You want 1.6*10^3*D*v = 1. using D= 0.07 I get 8.93*10^-3m/s

 

[Oblio]

b: I get (1.6 x 10^-4) x (7cm) = 11.2 x 10^-6

c: (.25) x (7cm^2) = 1.225 x 10^-3

 

[Oblio]

It matters what part of the equation you use?

 

[learningphysics]

It matters what part of the equation you use?

No. It's all the same. But it's probably most convenient to use 1.6*10^3 D*v, since you can just plug in the diameter.

 

[Oblio]

I must be doing something wrong with the other part...

 

[Oblio]

"For what approximate range of speeds is it safe to treat the drag force as purely quadratic".

ie when is linear negliable?

 

[learningphysics]

"For what approximate range of speeds is it safe to treat the drag force as purely quadratic".

ie when is linear negliable?

Not sure... it's a matter of opinion. When the ratio is 10 or more I'd say...

I think the reason our numbers are off are because gamma/beta = 1.5625*10^3... not 1.6*10^3

when I use 1.5625*10^3, I also get 9.14*10^-3m/s. So I think that's right.

 

[Oblio]

Ah, you that's right.

So I understand the question right, in that they want when the linear force is negligable?

 

[learningphysics]

Ah, you that's right.

So I understand the question right, in that they want when the linear force is negligable?

Yeah. Negligible compared to the quadratic term.

 

[Oblio]

Great.
I guess I can say that its ok to ignore the linear force under normal conditions since the quadratic is so much more.

 

[learningphysics]

Great.
I guess I can say that its ok to ignore the linear force under normal conditions since the quadratic is so much more.

Yes, I agree. For a ratio of 10 or more (ie quadratic is 10 times linear force), the velocity needs to be at least 0.0914m/s. The baseball will be going much faster than that, so you can definitely ignore the linear term.