# Projectile with Friction

I am trying to find the optimal angle for a projectile to travel a maximum horizontal distance.

I've been given the projection has mass $1kg$ and initial velocity $150m/s$. The equation for air resistance I'm using is $$F_{air} = - k|v|^2$$

where $k = 0.001$

I'm currently trying to use the following equation to find the horizontal distance:

$$\int v_x dt = \int\frac{1}{\frac{1}{v_o\cos(\theta)} + \frac{kt}{m}} dt$$

but I'm unsure what values to integrate between on the right integral as I do not know the time it will take the projectile to land.

Please can someone give me a hint on what I should do next.

• If the ground is flat, you need to solve for $y(t)=0$. – Yves Daoust Apr 20 '16 at 9:11
• @YvesDaoust Yes, the ground is flat. Thank you – Nique Apr 20 '16 at 9:12
• @YvesDaoust So $\int v_x dt = 0$ and then rearrange to find $\theta$? – Nique Apr 20 '16 at 9:19
• No. $\int v_y\,dt=0$ ! – Yves Daoust Apr 20 '16 at 9:21
• @YvesDaoust But I have no equations involving $v_y$?? – Nique Apr 20 '16 at 10:57