Finding Range of Projectile Under Air Resistance

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Discussion Overview

The discussion centers on the challenge of determining the range of a projectile under the influence of air resistance. Participants explore the complexities involved in obtaining a formula for the maximum distance traveled in the x direction, particularly when traditional analytical methods become inapplicable due to the presence of air resistance.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes their efforts to plot the projectile's path using Matlab and expresses difficulty in deriving an equation for the range due to air resistance complicating the time-of-flight calculation.
  • Another participant notes that solving for the x and y displacements requires numerical methods, indicating that analytical solutions are not feasible, which complicates finding the range.
  • A question is raised regarding whether air resistance is proportional to velocity, suggesting a potential area for further exploration in modeling the projectile's motion.
  • A suggestion is made to utilize a symbolic toolbox for a quasi-analytic solution or to simplify the model to a more basic form to facilitate calculations.

Areas of Agreement / Disagreement

Participants generally agree on the challenges posed by air resistance in obtaining a range equation, but multiple approaches and techniques are proposed without consensus on a specific method or solution.

Contextual Notes

The discussion highlights limitations in analytical methods due to the complexity introduced by air resistance, and the need for numerical techniques to approximate solutions. There is also uncertainty regarding the nature of air resistance and its impact on the projectile's motion.

Who May Find This Useful

This discussion may be of interest to those working on projectile motion problems in physics, particularly in contexts involving air resistance, as well as individuals using numerical methods in computational modeling.

sosolid
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I have made a program in Matlab to plot the path of a projectile with air resistance, however, I am lost as to how to obtain an equation for the range. I have found my displacements in the x and y direction by solving second order differential equations, it would be far too difficult to set the y equation to zero and hence solve for the time. Basically I want a way of obtaining a formula to show the maximum distance traveled by the projectile in the x direction. If the projectile did not have air resistance then this would be easy as one would simply have to set the equation for vertical height to zero and solve for the time, this time is the total time of flight. However, when considering air resistance I am just finding it too difficult.
 
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sosolid said:
I have made a program in Matlab to plot the path of a projectile with air resistance, however, I am lost as to how to obtain an equation for the range. I have found my displacements in the x and y direction by solving second order differential equations, it would be far too difficult to set the y equation to zero and hence solve for the time. Basically I want a way of obtaining a formula to show the maximum distance traveled by the projectile in the x direction. If the projectile did not have air resistance then this would be easy as one would simply have to set the equation for vertical height to zero and solve for the time, this time is the total time of flight. However, when considering air resistance I am just finding it too difficult.

Maybe I'm missing something here.

You have already seen that to solve for x and y, you have to solve the differential equation numerically, i.e. there are no analytical solutions to each of the coordinate position. This tells you that you also can't solve for the time-of-flight analytically, and that means that you also have the same issue with solving for the range.

So unless you want to set up 2 coupled differential equations, the only way I can see in doing this is to solve for the range separately using your original equation that solved for x and y, but doing it in smaller steps until you narrow down the solution for x corresponding to y~0. There are several numerical techniques that may be available in Matlab to find such a root.

Zz.
 
Is air resistance proportional to the velocity?
 
Hi,
Well, if you have a symbolic toolbox, you might try to get a quasi-analytic solution.
Or at least try to reduce your model to something more basic.
Smoochie
 

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