- #1

Hector Triana

- 7

- 2

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.

Thanks for your attention.