Varying Gravity and Air Resistance in projectile motion

In summary, the conversation discussed the problem of projectile motion considering the variation of gravity acceleration and air resistance. The mathematical model for rising and falling of the ball was questioned, and a two-dimensional problem was proposed where the acceleration of gravity is constant near the surface of the Earth and air resistance is ignored. However, if air resistance is included, differential equations for the horizontal and vertical directions should be written. It was also suggested to consider the effects of the centrifugal and Coriolis forces, as well as the curvature of the Earth. The use of numerical methods and simulations was also mentioned as an approach to solving this problem. The speaker also shared their experience in using computer programs to simulate the problem. Finally, it was advised to start with a
  • #1
Hector Triana
7
2
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.
 
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  • #2
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.
 
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  • #3
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.
 
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  • #4
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.
 
  • #5
Hector Triana said:
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.
 
  • #6
sophiecentaur said:
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.
 
  • #7
Also note that air resistance more properly is proportional to ##v^2## in a case such as this.
 
  • #8
boneh3ad said:
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.
 

Related to Varying Gravity and Air Resistance in projectile motion

1. How does varying gravity affect projectile motion?

The force of gravity is what causes a projectile to accelerate downward. Therefore, the greater the gravitational force, the faster the projectile will accelerate and the shorter the flight time will be. Conversely, reducing gravity will result in a slower acceleration and a longer flight time.

2. What is the relationship between air resistance and projectile motion?

Air resistance, also known as drag, is the force that opposes the motion of a projectile through the air. As a projectile moves faster, the amount of air resistance it experiences increases. This can cause a decrease in its velocity and a change in its trajectory.

3. How do varying air densities affect projectile motion?

The density of air can vary depending on factors such as altitude and temperature. In general, denser air will result in a greater amount of air resistance and a shorter flight time for a projectile. This is because the air molecules are more tightly packed, creating more resistance for the projectile to overcome.

4. Can varying gravity and air resistance affect the range of a projectile?

Yes, the range of a projectile is affected by both gravity and air resistance. A higher gravitational force will result in a shorter range, while air resistance can also decrease the range by slowing down the projectile's velocity. In some cases, adjusting these factors can help increase the range of a projectile.

5. How can we calculate the effects of varying gravity and air resistance on projectile motion?

There are various equations and mathematical models that can be used to predict the effects of varying gravity and air resistance on projectile motion. These include the equations for calculating gravitational force and air resistance force, as well as the equations for determining the trajectory and range of a projectile. These calculations can be complex and often require advanced mathematical skills.

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