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I am looking for a game that would be suitable for teaching basic orbital mechanics concepts to primary school students (grades 6-8). Contrary to most of the questions here asking about spaceflight simulators and such, I am not looking for super realistic representations of orbital mechanics. I want to allow students to tinker around with basic central force motion and see the ways in which conic sections are altered by thrust, etc. Seeing/enacting an example of rendezvous (maybe in a CW frame?) would be neat too.

I think Kerbal Space Program is going to be a common answer, and it is suitable, but unfortunately the fact that it is commercial makes it tougher to use in an educational setting. Extra points if there is something browser based, but my hopes aren't high for that.

UC Boulder has a project, PhET, that has many free, interactive, in-browser math and science modules. They have one called Gravity and Orbits that's written in HTML5, making it compatible with most modern browsers (including Safari on iPads). If you click the 'For Teachers' drop-down, you'll find it even has quite a few resources for lesson planning with it for middle schoolers.

I think this will suite your needs of it being appropriate for a class room and browser-friendly (and free!).

As a side note, if you want to tie Gravity and Orbits, or other software you end up choosing, to how the entire Solar System works, NASA has a great interactive site called Solar System. If you click the concentric circles on the menu bar, you can see the planets (and more) orbiting our Sun. You can also zoom in and out like if it was Google Maps. You can click a planet (or other object) to find out more information on it.

Personally, I teach orbital mechanics classes to preschoolers, elementary and middle school kids using a makeshift trampoline with stretchable cloth clamped to the rim. Place a heavy weight (e.g. a dumbell) in the middle to simulate a large massive body like the earth or the sun. Use marbles to illustrate a spacecraft or planets. You can easily show the basic relation between distance and speed in an orbit or trajectory. Giving the marbles a push while in “orbit” shows how impulse thrusts affect the shape of the orbit too.

See this YouTube video for other ideas you can illustrate using this kind of demo

They can be easily constructed using PVC pipes. See these instructions for constructing your own gravity well. In my experience, the larger the well, the easier it is to demonstrate orbital mechanics.

I want to allow students to tinker around with basic central force motion and see the ways in which conic sections are altered by thrust, etc. Seeing/enacting an example of rendezvous (maybe in a CW frame?) would be neat too.

I definitely think KSP is the right answer here. The ways in which it departs from real-world space flight (such as a much smaller home planet, which leads to ascent to low orbit taking ~3 minutes instead of ~10 minutes) make sense for the classroom.

the fact that it is commercial makes it tougher to use in an educational setting.

The KerbalEDU version is offered with an educator discount.

The free option is Orbiter, which is more realistic in some ways than KSP, but less user friendly.