From projectile motion to orbits using Geogebra

I was inspired last night while watching the launch of the Mars Science Laboratory that instead of doing banked curve problems (which are cool, but take a considerable investment of algebra to get into) we would move on to investigating gravity.

The thing that took me a long time to wrap my head around when I first studied physics in high school was how a projectile really could end up orbiting the Earth. The famous Newton drawing of the cannon with successively higher launch velocities made sense. I just couldn’t picture what the transition looked like. Parabolas and circles (and ellipses for that matter) are fundamentally different shapes, and at the time the fact that they were all conic sections was too abstract of a concept for me. Eventually I just accepted that if you shoot a projectile fast enough tangentially to the surface of the Earth, it would never land, but I wanted to see it.

Fast forward to this afternoon and my old friend Geogebra. There had to be a way to give my physics students a chance to play with this and perhaps discover the concept of orbits without my telling them about it first.

You can download the sketch I put together here.

The images below are the sorts of things I am hoping my students will figure out tomorrow. From projectile motion:

…to the idea that it is still projectile motion when viewed along with the curvature of the planet:

Continuing to adjust the values yields interesting results that suggest the possibility of how an object might orbit the Earth.

If you open the file, you can look at the spreadsheet view to see how this was put together. This uses Newton’s Law of Gravitation and Euler’s method to calculate the trajectory.You can also change values of the variable deltat to predict movement of the projectile over longer time intervals. There is no meaning to the values of m, v0, or height – thankfully the laws of nature don’t care about units.

As is always the case, feel free to use and adjust this, as well as make it better. My only request – let me know what you do with it!