# Projectile help

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 travelled 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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ZapperZ
Staff Emeritus
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 travelled 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