# How can I find the initial velocity of a projectile from its range, when accounting for air resistance?

I'm trying to find the initial velocity of a projectile based on it's range. I know how to use kinematic equations to do this when air resistance is ignored, but I'm not sure how to take into account air resistance.

Given:

• initial position $$(0, h)$$
• a projectile with a drag force $$D(v^2)$$, mass $$m$$
• A range $$h$$
• Gravitational acceleration $$g$$

I want to find an initial velocity:

• $$(0, v_i)$$

I've found a number of similar problems, but they all either treat drag force as a function of $$v$$ rather than $$v^2$$, or find the range from the initial velocity.

Either an analytical or numeric solution would work.

• This problem is more difficult than it seems. Drag force is proportional to velocity squared. It is also proportional to the density of the fluid that it is moving through, and the density of air varies with altitude. Drag force is also proportional to the drag coefficient and the cross-sectional area that is perpendicular to the flight path, which will both vary if the projectile is an odd shape and tumbling as it flies through the air. Go with the numeric solution. – David White Sep 13 at 5:16
• @DavidWhite The object is a sphere in this case so rotation/changing direction isn't an issue. Could you give me some ideas on how to model it numerically? – Zac Pullar-Strecker Sep 14 at 8:55