# Air Resistance Problem

• Dec 15th 2007, 06:42 AM
bobak
Air Resistance Problem
I found a question to find an expression for the maximum height that a particle projected vertically with a velocity of u can reach if there is a resistive force acting on it of Kmv newton (where m is the particles mass, v being it velocity and k is a constant) and gravity acting on it as well.

after doing a bit of algebra/calculus i got an answer of

. . . . . $\displaystyle \boxed{ \frac{g}{k^2} ln \left(\frac{g}{ku+g} \right) + \frac{u}{k} }$

What i notice i that my solution i not defined for the case k=0, meaning there is a clearly insonsistent with the solution obtained when no air resistance is accounted for. So is my solution incorrect ?
• Dec 16th 2007, 11:12 AM
topsquark
Quote:

Originally Posted by bobak
I found a question to find an expression for the maximum height that a particle projected vertically with a velocity of u can reach if there is a resistive force acting on it of Kmv newton (where m is the particles mass, v being it velocity and k is a constant) and gravity acting on it as well.

after doing a bit of algebra/calculus i got an answer of

. . . . . $\displaystyle \boxed{ \frac{g}{k^2} ln \left(\frac{g}{ku+g} \right) + \frac{u}{k} }$

What i notice i that my solution i not defined for the case k=0, meaning there is a clearly insonsistent with the solution obtained when no air resistance is accounted for. So is my solution incorrect ?

Your solution is correct. (Good job, by the way. :) This is a long problem.) What you need to do in some cases (I don't know how to predict which) is go all the way back to your initial motion equation and set k = 0. The mere presence of the resistance term changes the character of the process by which the solution is obtained. Apparently enough so that the nature of the solution changes.

Since this is a physical equation, after all, it is likely that as k tends to 0, the value of the maximum height will approach the k = 0 value.

-Dan
• Dec 16th 2007, 12:31 PM
bobak
Quote:

Originally Posted by topsquark
(This is a long problem.)

Yeah i was thinking about posting my working but it would have taken too long.

the solution obtained when no air resistance is accounted for is $\displaystyle \frac{u^2}{2g}$

I tested the equation using k=0.01 and the two solutions agreed within 2d.p. then i used the taylor series for lnx and showed that for small values of k the solution obtained as k apporached zero is equal to the other solution.

gosh i have never use so much pure math in a mechanics problem.
• Dec 17th 2007, 04:37 AM
topsquark
Quote:

Originally Posted by bobak
gosh i have never use so much pure math in a mechanics problem.

If you think that's bad, try taking graduate E and M! :D

-Dan