Simple Newtons laws question ?

[sid9221]

"Simple" Newtons laws question ?

At time t = 0 a particle of unit mass is projected vertically upward with velocity v_0 against gravity, and the resistance of the air to the particle's motion is κ times its velocity.
Show that during its flight the velocity v of the particle at time t is

$$v = (v_0 +\frac{g}{k}) e^{-kt} - \frac{g}{k}$$

Now what I have done is work with Newtons 2nd law:

$$mx'' = -mκv$$
$$v' = -κv$$

solving this differential equation gives:

$$v=Ce^{-kt}$$

Now v(0) = v_0 from question so:

C=v_0

$$v=v_0 e^{-kt}$$

Now clearly I missing quite a bit of stuff, so where exactly am I going wrong ?

 

[LCKurtz]

You have left the force of gravity out of the differential equation.