Determining the maximum braking power using derivations

[arhzz]

Homework Statement

The braking force K: [0, ∞) → R of an eddy is a function of velocity given by $\frac{v}{v^2+9}$At what speed does the braking force reach its greatest value?

Relevant Equations

Differantiation

Hello! So what I've tried to tackle this problem is derive the equation,set it equal to zero,find a value for v and than put it in the second derivation.So when I derive this I get $$ \frac{v^2+9-v *(2v)}{v^4+8v^2+16)} $$ Now if i set that equal 0 and try to find a value for v I get this.

$-v^2 + 9 = 0$ and now we have to values 3 and -3 Now if I do the second derivation I get this $$ \frac{4v*(v^2-9)}{(v^2+9)^3} - \frac{2v}{(v^2+9)^2} $$

Now if i input the value of 3 in I get -0,0185.And for - 3 the same but without the minus. so 0,0185 Now I'm not really sure this is right I'd reckon if it is only the value of -3 can be right because velocity can't be negative as far as I know.But do you reckon my method is correct or maybe I had a slip up in the derivation. Thanks!

 

[kuruman]

Homework Statement:: The braking force K: [0, ∞) → R of an eddy is a function of velocity given by $\frac{v}{v^2+9}$At what speed does the braking force reach its greatest value?
Relevant Equations:: Differantiation

ello! So what I've tried to tackle this problem is derive the equation,set it equal to zero,find a value for v and than put it in the second derivation.

If you want to maximize F with respect to v, you set $\dfrac{dF}{dv}=0$, solve for v then put the value of v you found back into the expression for F. What do you mean by "second derivation"?

Note that F is a braking force and must be opposite to the velocity. If v is positive, F must be negative and vice-versa.

 

[arhzz]

If you want to maximize F with respect to v, you set $\dfrac{dF}{dv}=0$, solve for v then put the value of v you found back into the expression for F. What do you mean by "second derivation"?

Note that F is a braking force and must be opposite to the velocity. If v is positive, F must be negative and vice-versa.

Oh okay,what I had in mind is to do it like in mathematics with when dealing with extrem of a function.But I'll try it this way.

 

[kuruman]

Oh okay,what I had in mind is to do it like in mathematics with when dealing with extrem of a function.But I'll try it this way.

This is mathematics. What you probably had in mind is taking the second derivative in order to determine whether the extremum is a maximum or a minimum.

 

[arhzz]

This is mathematics. What you probably had in mind is taking the second derivative in order to determine whether the extremum is a maximum or a minimum.

Exactly that! I just didnt know how to say it in english.Because I though the maximum should be the solution,I guessed since it said "when does it reach its greatest value" I assumed the maximum was the thing I sought.I've done as you have suggested and when I input -3 in K (or F as you say it) I get K(-3) = -0,16777

 

[kuruman]

Exactly that! I just didnt know how to say it in english.Because I though the maximum should be the solution,I guessed since it said "when does it reach its greatest value" I assumed the maximum was the thing I sought.I've done as you have suggested and when I input -3 in K (or F as you say it) I get K(-3) = -0,16777

That sounds right.

 

[arhzz]

That sounds right.

Perfect,thank you!