# Finding the motion of a projectile under the action of drag forces using integration

So I am considering a projectile fired with velocity components $U_x$ and $U_y$ from position $(0,0)$ at time $t=0$ with drag forces proportional to the square of the speed in each direction (x and y directions) so I let the drag force in the x direction be $-cV_x^2$ and in the y direction be $-bV_y^2$. My first question is:

• Are these drag forces typical of projectile motion through air only? Or, for the case in air, is dray force simply proportional to the speed (not the square)? If so, what are examples of where the resistive force would be proportional to the square of the speed?

Now my main problem is with signs. For motion in the x direction, the situation is simple as the net acceleration is

$a_{net}=\frac{-bV_x^2}{m}$

which can be integrated simply using separation of variables.

However when I am considering the y direction, I noticed that saying

$F_{drag}=-cV_y^2$

is wrong because $V_y^2$ is a positive quuantity, so then my frictional force does not always act in the opposite direction to motion. I would instead have to make

$F_{drag}=-c\mid V_y\mid V_y$

But then integrating the acceleration to find velocity would require integration of a term proportional to

$\int \frac{1}{\mid V_y\mid V_y}dV_y$

which I don't know how to approach. Is there a better way of doing this to get the correct signs?

• You could do separate integrals for $V_y<0$ and $V_y>0$. – sammy gerbil Dec 10 '16 at 5:43