Varying Gravity and Air Resistance in projectile motion

Salutations, I have been trying to approach a case about projectile motion considering variation of gravity acceleration and air resistance:

A spherical baseball with mass "m" is hit with inclination angle $$\theta$$ and launching velocity $$v_0$$, then, the wind has a drag force equals to $$F=kv$$ and according the acceleration of gravity force is varying in function of height.

So, analzying the gravity in function of height, I got this: $$mg=\frac{GM_Tm}{\left(R+y\right)^2}\\\ \\ g=\frac{GM_T}{R^2 \left(1+\frac{y}{R}\right)^2}\\\ g=\frac{g_+}{\left(1+\frac{y}{R}\right)^2}$$

Then, regarding the gravitational varying acceleration according height of the ball is considered insignificant above the radius of the Earth, which I considered to apply binomial expansion $$((1+x)^n=1+nx$$): $$\Gamma=g\left(1+\frac{y}{R}\right)^{-2}\\\ \Gamma=g\left(1-\frac{2y}{R}\right)$$

After that, analyzing the applied forces to the ball when rises up, I got this equation: $$ma=-m\Gamma-kv\\ \ a=-\Gamma-\frac{k}{m}v\\ \frac{dv}{dt}=-g\left(1-\frac{2y}{R}\right)-\frac{k}{m}v\\$$

So, when the ball is falling down, I consider this model: $$\frac{dv}{dt}=\frac{k}{m}v-\Gamma\\ \frac{dv}{dt}=\frac{k}{m}v-g\left(1-\frac{2y}{R}\right)$$

The objective of the modelling is finding maximum height, total flight time of the ball and maximum horizontal displacement

Finally, my doubt is: -Are the mathematical model well posed for rising and falling down of the ball?

This is just academic curiosity, and it's the first time that I approach varying gravity and air resistance in projectile motion, and I'm not sure if the varying gravity is well applied in the models.

So, I would like any guidance or starting steps or explanations to find the solutions because it's an interesting case of projectile motion.

• Looks good to me, as long as your equations of motion are correct. Seems like you accounted for everything relevant. – InertialObserver Dec 24 '18 at 1:11
• You don't need seperate models for upwards and downwards motion, if you consider $v$ to be positive when moving upwards and negative when moving downwards then only one model is required. – Eddy Dec 24 '18 at 15:03
• Hi, thanks for your commentaries, @InertialObserver, these equations seems correct, but the factor $\Gamma=g\left(1-\frac{2y}{R}\right)$ makes me confused when I try to find maximum height about how to manipulate $y$ in differential equation or numerical methods with initial conditions are needed? – ht1204 Dec 24 '18 at 17:19
• Hi, @Eddy, regarding models, what mathematical model do you suggest? Especially when I try to find maximum height about to approach the factor $g\left(1-\frac{2y}{R}\right)$ caused by varying gravity, I got a procedure and an answer, but Im not convinced and I saw case when the ball was analyzed in moving up and falling down and I saw two models, that's the reason of my doubt. – ht1204 Dec 24 '18 at 17:27
• I think your model for the ball rising up should be fine for both up and down. If you want to improve it then use the full equation for gravity (not just the linear expansion), or investigate better approaches to modeling air resistance. – Eddy Dec 24 '18 at 17:53