sketch the parametric curve $x = 25t − 30 \sin t$ and $y = 75 − 30 \cos t$

Question

The motion of a particle is defined by the equations $x = 25t − 30 \sin t$ and $y = 75 − 30 \cos t$, where $x$ and $y$ are expressed in millimeters and $t$ is expressed in seconds.

Sketch the path of the particle for the time interval $0 < t < 2\pi$ and determine... (how to sketch this?)

Answer 1

HINT

to help your imagination:

Answer 2

As it is a cinematics issue (motion of a particle), it is possible to give a geometrical understanding, as a superimposition of a rotational effect to a translational effect (see figure below), due to the fact that the initial parametric equations:

$$x = 25t − 30 \sin t, \ \ \ \ y = 75 − 30 \cos t$$

can be written under the vector form:

$$\begin{pmatrix}z\\y\end{pmatrix}=\underbrace{\begin{pmatrix}25t\\75\end{pmatrix}}_{\text{translational}}-\underbrace{30\begin{pmatrix}\sin(t)\\ \cos(t)\end{pmatrix}}_{\text{rotational}}$$

giving rise to the following picture (imagine a pedestrian walking horizontally on a bridge at speed 25 with a rotating ball at the end of a rope of length $30...$).

Remark: you are asked the part of this curve corresponding to $[0,2\pi]$ (as drawn by @Zoli).

Answer 3

Here a plot: Its is an arc of trochoid, which looks like this on a longer interval: