Varying Gravity and Air Resistance in projectile motion

[Hector Triana]

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.

Thanks for your attention.

 

[kuruman]

This is a two dimensional problem because the motion takes place in a plane. The usual approximation is that the acceleration of gravity is constant near the surface of the Earth and that air resistance can be ignored. Then one writes two differential equations for the horizontal and vertical directions:$$\frac{dv_x}{dt}=0~;~~\frac{dv_y}{dt}=-g$$where "up" is positive and "down" is negative. If you want to add air resistance that is proportional to the velocity and include an altitude-dependent acceleration of gravity, then you should write $$\frac{dv_x}{dt}=-bv_x~;~~\frac{dv_y}{dt}=-g(y)-bv_y$$for a projectile moving up and to the right and $$\frac{dv_x}{dt}=-bv_x~;~~\frac{dv_y}{dt}=g(y)-bv_y$$for a projectile moving down and to the right. Note that the positive x and y axes are defined by the direction of the velocity. Strictly speaking, if you worry about the variation of $g$ with altitude, you should also worry about the curvature of the Earth in which case it would be more correct to formulate the problem in spherical rather than Cartesian coordinates and start considering the effects of the centrifugal and Coriolis forces at the latitude where the projectile is launched.

 

[sophiecentaur]

Doing this analytically could be a problem but it would not be hard to use numerical methods. The formulae are easy to implement with a computer program - step by step calculations and you can vary the g dependence with height and the contribution of air resistance. It's how the simulation programs do it and, if you happen to be happy with coding, it can be entertaining and instructive. (I know that purists may say "so what?" but simulations are very much in fashion these days.)
I have done this myself in the past using the Psion programming language and Visual Basic inside Excel. With faster machines, these days, you could do it fast with very small steps so the errors would be small.

 

[Hector Triana]

Hello, thanks for your commentaries, I'm looking for approaches for this studying case, especially without initial conditions, @sophiecentaur , would you share the simulation in excel? it sounds interesting, that point of view seems very analytical. The objective that I've got with this exercise is how to model it and solve it with differential equations step by step. Thanks for your attention.

 

[sophiecentaur]

would you share the simulation in excel?

Sorry but that was long ago and I have no copy of it. In 2D, it's just a matter of calculating the forces on the object every Δt interval, starting at a given x,y and initial Vx and Vy. It could be worth while starting with a simple uniform g, flat Earth and no friction and then introduce more factors.

 

[kuruman]

It could be worth while starting with a simple uniform g, flat Earth and no friction and then introduce more factors.

And if you get stuck, we are here to help. I think you will learn a lot when you do it yourself from scratch rather than being influenced by how someone else has done it.

 

[boneh3ad]

Also note that air resistance more properly is proportional to $v^2$ in a case such as this.

 

[sophiecentaur]

Also note that air resistance more properly is proportional to $v^2$ in a case such as this.

At the flick of a switch, you can change that in the coding. Good fun but can turn you into a coding nerd.