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Equations of Motion for a Projectile in Polar Coordinates
Why? There is no why.
Suppose a ball is shot at an angle θ with an initial velocity v 0. What will the motion be like in polar coordinates?
First, let me start with Newton’s 2nd Law in polar coordinates.
Where do those come from? Here is my derivation of the acceleration in polar coordinates. For fun.
Once the ball is in the air (and ignoring air resistance) the only force on the ball is the gravitational force. Yes, this would be -mg in the y-direction, but we don’t have a y-direction. Instead, we have polar coordinates. Maybe this picture will help.
The r and θ components of the gravitational force will change as:
If I use these forces with Newton’s law in polar coordinates, I get:
Of course the mass cancels — but now I can solve the first equation for (r-double dot) and the second equation for (theta-double dot).
It doesn’t matter that these second derivatives depend on the other stuff — I can still…
