# Conservation of energy problem (projectile with air drag)

1. Nov 1, 2008

### azure kitsune

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

work done by an external force = change in energy

3. The attempt at a solution

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

\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*}

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

We can set the two expressions equal

\begin{align*} -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*}

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

2. Nov 1, 2008

### tiny-tim

Hi azure kitsune!

Only the last line is wrong.

3. Nov 2, 2008

### azure kitsune

Waaahhhh!!! *feels so stupid!!!*

Thanks tiny-tim.

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