A particle of mass $\displaystyle m$ is projected with speed $\displaystyle u$ along a straight horizontal track. The first section of the track has length $\displaystyle d$. On this section of the track the motion is resisted by a

constant force of magnitude $\displaystyle mk$, where $\displaystyle k$ is a positive constant. The particle does not come to rest on this first section of the track.

Show that the speed $\displaystyle V$ of the particle at the end of the first section of the track is given by:

$\displaystyle V = \sqrt{u^2 - 2kd}$

Using the equation $\displaystyle ma = F$, I got the following equation.

If we let the speed of the particle = $\displaystyle v$

$\displaystyle m\frac{dv}{dt} = v-mk$, however in the solutions they just have $\displaystyle m\frac{dv}{dt} = -mk$, what's happened to the $\displaystyle v$, I would have thought that the forces acting on the paticle would be $\displaystyle v$ and $\displaystyle -mk$?

Thanks for the help