Rotating Stick

The animation at right shows the basic animation - a stick of length 1 rotating about the origin. We name the endpoints A (orange) and B (purple) and so the stick is

(1 - t)A + tB , {0 ≤𝑡≤1}

Then with 𝑇 {0 ≤ 𝑇 ≤ 360} we rotate the two points about the origin at a rate of one degree per second. We’re going to call each time unit in Desmos a second for convenience, even though they aren’t precisely seconds.

A = (cos⁡(𝑇), sin(𝑇)) and B = (cos⁡(180 + 𝑇), sin(180 + 𝑇))

As the points rotate, they carry the stick along with them.

Desmos Tip: On the animations we share, if you click on the bottom right corner of the graph you'll be able to see and play with the graph in Desmos in a new windown.

Parabola Stick

In this activity it’s time to start moving the centre of rotation. To mimic throwing the stick through the air we begin at the origin at an angle of 45°, and have it rotate clockwise with constant horizontal velocity, and vertical velocity under the force of gravity. Given that, the centre of the stick follows a parabola that we take to be:

y = p(x) = 18x(16 - x) {0 ≤ x ≤ 16}

We choose the horizontal speed to be 1/s and the rotational speed to be 360°/s. Then the A (orange) and B (purple) dynamic equations are:

A = (T + cos⁡(45 - 360T, p(T) + sin(45 - 360T))

B = (T + cos⁡(225 - 360T, p(T) + sin(225 - 360T))

Give the animation a try, and click the “edit graph on desmos” button in the bottom right corner of the animation when you’re ready to see how we’ve done it here.