Conservation of energy problem (projectile with air drag)

[azure kitsune]

Homework Statement

A stone with weight w is thrown vertically upward into the air from ground level with initial speed v₀. If a constant force f due to air drag acts on the stone throughout its flight, (a) show that the maximum height reached by the stone is

h = \frac{v_0^2}{2g(1+f/w)}

(b) Show that the stone's speed is

v = v_0 \left( \frac{w-f}{w+f} \right) ^ {1/2}

just before impact with the ground.

Homework Equations

work done by an external force = change in energy

The Attempt at a Solution

I have no trouble with part (a). I need help with (b)

\begin{align*}<br /> W_{air} &amp;= (-f) * (2h) \\<br /> &amp;= -2fh \\<br /> &amp;= -2f\frac{v_0^2}{2g(1+f/w)} \\<br /> &amp; = -2f\frac{v_0^2 w}{2g(w+f)} <br /> \end{align*}

W_{air} = \Delta E = \Delta K = \frac{1}{2}m(v^2 - v_0^2) = \frac{w}{2g}(v^2 - v_0^2)

We can set the two expressions equal

\begin{align*}<br /> -2f\frac{v_0^2 w}{2g(w+f)} &amp;= \frac{w}{2g}(v^2 - v_0^2) \\<br /> -2f\frac{v_0^2 }{(w+f)} &amp;= v^2 - v_0^2 \\ <br /> v^2 &amp;= v_0^2 + -2f\frac{v_0^2 }{w+f} \\<br /> &amp;= v_0^2 \left( \frac{1-2f}{w+f} \right)<br /> \end{align*}

But this is wrong. Can anyone tell me where I messed up?

 

[tiny-tim]

\begin{align*}<br /> v^2 &amp;= v_0^2 + -2f\frac{v_0^2 }{w+f} \\<br /> &amp;= v_0^2 \left( \frac{1-2f}{w+f} \right)<br /> \end{align*}

But this is wrong. Can anyone tell me where I messed up?

Hi azure kitsune!

Only the last line is wrong.

 

[azure kitsune]

Waaahhhh! *feels so stupid!*

Thanks tiny-tim.