# Calculating the angle an object has to be launched at with simulated drag [closed]

I’m trying to calculate the angle(s) a projectile has to be launched at in a game that has a simulated drag which reduces the velocity by 2.5% of the current velocity per tick (25 milliseconds). This is independent from it’s shape and mass.

I'm given initial speed, distance and gravity. I know that theta is:

Where $$v$$ is the initial velocity, $$x$$ the distance and $$y$$ the initial height.

And I figured that the function of velocity for an object after it’s launched is exponentially decaying so I came up with this function:

Where v_0 is the initial speed, 2.5 is the percentage and 0.025 the tick rate.

Is there a way to account for this drag in the original formula or is there a different one? So far I only found formulas for linear drag but I'm not sure how I would incorporate this as this drag isn’t dependent on an objects mass and terminal velocity.

• Where is the drag coefficient in your equation?
– Eli
Dec 5, 2022 at 15:47

What you're doing is modeling an object with "linear drag". Since the rate at which the velocity decreases is proportional to the velocity itself, we can write $$\frac{\Delta v_x}{\Delta t} = - \frac{0.025 v_x}{25 \text{ ms}} = - \alpha v_x$$ (more or less), with $$\alpha = 10 \text{ s}^{-1}$$. A similar equation holds for $$v_y$$, but with the gravitational acceleration of $$-g$$ factored in.
Since the drag is linear, it is possible to solve exactly for the position and the velocity as a function of time; I did so in my answer here. However, finding a "range equation" is a much harder; you would have to find the time $$t$$ of the landing of the object and then figure out the value of $$x$$ at that time, which I don't think is actually doable algebraically (since $$y(t)$$ is transcendental in $$t$$.) Trying to find a nice function for $$\theta$$ as a function of the desired range is probably a fool's errand.