A rubber ball gets thrown horizontally from $H$ height find the horizontal distance of the ball till it hits the ground
There's a rubber ball that has mass $m$, it gets thrown horizontally (from $H>0$ height) with initial velocity of $\vec V(0) = V_0 \hat X$ ($\hat X$ is the axis, meaning horizontal) in addition to the gravity, there's wind resistance given by $\vec f(\vec V)=-b \cdot \vec V$ 1) find the horizontal and vertical velocity equations 2) find the horizontal distance of the ball till it hits the ground ( first time it hits the ground)
My try: for the horizontal axis $(\hat X)$ according to <PERSON>'s second law of motion $\sum F_x = m \cdot a_x$ $\to$ $-b \cdot V_x=m \cdot a_x$ $\to$ $a_x= \dot V_x= \frac{-b \cdot V_x}{m}$
And for the vertical axis $(\hat Y)$ $\sum F_y = m \cdot a_y$ $\to$ $-b \cdot V_y -m \cdot g =m \cdot a_y$ $\to$ $a_y = \dot V_y= \frac{-b \cdot V_y}{m} - g$
this is for the first part.
My try for the second part:
Solve the $ODE$ for both horizontal and vertical in order to find the $V_y(t)$ and $V_x(t)$ $$\dot V_y(t) = \frac{-b \cdot V_y}{m} - g $$ $\to$ $$V_y(t) = A \cdot e^{\frac{-b}{m} \cdot t}-\frac{m \cdot g}{b}$$
then we substitute $V_y(0)=A \cdot 1 - \frac{m \cdot g} {b} = 0$ $\to$ $A= \frac {m\cdot g} {b}$
then we substitute the $A$ we found $$V_y(t) = \frac {m\cdot g} {b} \cdot e^{\frac{-b}{m} \cdot t}-\frac{m \cdot g}{b}$$
now we solve the $ODE$ for the horizontal axis $\dot V_x= \frac{-b \cdot V_x}{m}$ $\to$ $V_x(t) = A \cdot e^{\frac{-b}{m} \cdot t}$ then we substitute $V_x(0)= V_{0_x}=A$ $\to$ $V_x(t) = V_{0_x} \cdot e^{\frac{-b}{m} \cdot t}$
after we found $V_x(t)$ and $V_y(t)$ what i tried to do is find the $y(t) = 0$ by doing an integral on $V_y(t)=V_y(0) + \int_0^t V_y(t) dt$ (and the answer is $y(t)=H - \frac {g \cdot m} {b} \cdot (\frac {m}{b} \cdot e^{\frac {-b}{m} \cdot t } +t)$ in order to find the time it did hit the ground and then use that time in the horizontal distance equation to find that distance when it first hit the ground
but I got stuck and I do not know if this way is even correct as I could not continue from here
Sorry if there are some translation mistakes to English hopefully it is still understandable 512