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

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

(b) Show that the stone's speed is

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

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)

[tex]\begin{align*}
W_{air} &= (-f) * (2h) \\
&= -2fh \\
&= -2f\frac{v_0^2}{2g(1+f/w)} \\
& = -2f\frac{v_0^2 w}{2g(w+f)}
\end{align*}[/tex]

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

We can set the two expressions equal

[tex]\begin{align*}
-2f\frac{v_0^2 w}{2g(w+f)} &= \frac{w}{2g}(v^2 - v_0^2) \\
-2f\frac{v_0^2 }{(w+f)} &= v^2 - v_0^2 \\
v^2 &= v_0^2 + -2f\frac{v_0^2 }{w+f} \\
&= v_0^2 \left( \frac{1-2f}{w+f} \right)
\end{align*}[/tex]

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

 

[tiny-tim]

[tex]\begin{align*}
v^2 &= v_0^2 + -2f\frac{v_0^2 }{w+f} \\
&= v_0^2 \left( \frac{1-2f}{w+f} \right)
\end{align*}[/tex]

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.