Projectile in Polar Coordinates

Why?  There is no why.

Suppose a ball is shot at an angle θ with an initial velocity v0.  What will the motion be like in polar coordinates?

First, let me start with Newton’s 2nd Law in polar coordinates (I derived this in class).

F_r = m(\ddot{r}-r\dot{\theta}^2)
F_\theta = m(r\ddot{\theta}+2\dot{r}\dot{\theta})

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.

Photo - Google Photos 2018-02-18 08-55-07.png

The r and θ components of the gravitational force will change as:

F_r = -mg\sin\theta
F_\theta = -mg\cos\theta

If I use these forces with Newton’s law in polar coordinates, I get:

F_r = -mg\sin\theta=m(\ddot{r}-r\dot{\theta}^2)
F_\theta = -mg\cos\theta = m(r\ddot{\theta}+2\dot{r}\dot{\theta})

Of course the mass cancels – but now I can solve the first equation for \ddot{r} and the second equation for \ddot{\theta}.

\ddot{r} = r\dot{\theta}^2-g\sin\theta
\ddot{\theta} = \frac{-2\dot{r}\dot{\theta}}{r} - \frac{g\cos\theta}{r}

It doesn’t matter that these second derivatives depend on the other stuff – I can still calculate them. Once I have those, I can create a numerical calculation to update the velocities and positions. Suppose I have everything know at time(1), then the stuff at time(2) would be:

\dot{r}_2 = \dot{r}_1 +\ddot{r}\Delta t
\dot{\theta}_2 = \dot{\theta}_1 +\ddot{\theta}\Delta t
r_2 = r_1+\dot{r}\Delta t
\theta_2 = \theta_1 +\dot{\theta}\Delta t
t_2 = t_1 +\Delta t

Now I’m all set to do a numerical calculation. Well, I still need the initial conditions. I could use this:

\dot{r}_0 = v_0
r_0 =0
\dot{\theta}_0 = 0
\theta_0 = \theta_0

But wait! There’s a problem. The calculation for \ddot{\theta} has a 1/r term. If r is zero the universe will explode. I can fix this by having the initial r a little bit bigger than zero. Problem solved.

Here is the program in Glowscript.

For the first run, I am going to calculate the x- and y-coordinates in each step and plot x vs y.  I know this should look like a parabola.

newplot (2).png

Boom.  That works.  Now for a plot of both r and theta as a function of time for a high launch angle.

newplot (3).png

Double boom.

6 thoughts on “Projectile in Polar Coordinates

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