Given:
$x'' = -kx'*\sqrt{(x')^2+(y')^2}$
$y'' = -g -ky'*\sqrt{(x')^2+(y')^2}$
$x(0) = 0 , x'(0) = 20, y(0) = 0, y'(0) = 10,$ $k = 0.1, g=9.8$
Edit: Will I have to find the differential solutions first? Given what I know, the peak height will be when y' will be zero so I'm assuming a newton's method to find the root will work?
Using the Newton's Method, my thoughts are to go about it by finding the derivative and the maximum height will be when the velocity is 0 at a time other than 0 and when it's on the ground. Although I believe I could just take half of the time for the ball to hit the ground which would give me the time at maximum height as well.
Once I have the time that the ball will be at its maximum, it will hit the ground at double that height
Is this the correct way to go about this problem?
From the case with $k=0$ you can determine a first estimate for that time $T$. Then use numerical integration with the actual value of $k$ to get the state $(x,y,x',y')$ at $T$ and use $$ T_{next}=T-\frac{y(T)}{y'(T)} $$ Then either integrate from $0$ to $T_{next}$ or from $T$ to $T_{next}$ or from the last integration point where $y$ was positive and repeat this step with $T=T_{next}$ until $y(T)$ is small enough.
Note that with a step size $h$ the error in the state will be $O(h^4)$ and thus the error in $T$ will also always have a component of this size, even if the numerical value gives $y(T)=0$. Best would be to repeat the integration and end time determiniation with half the step size and take the difference as error estimate for the first value.