Question

19. "Describe the motion of a particle with position (`x,y`) as `t` varies in the given interval" ?

`x=5+2\cos\color(maroon)(cancel(t))\pit`, `y=3+2\sin\pit`, `1\let\le2`

*thanks to Ultrilliam for pointing it out,
`x=5+2\color(red)(\cos\pi\t)`

Answer 1

See explanation

Explanation:

`\cos\pit=(x-5)/2` and `\sin\pit=(y-3)/2`
`\therefore((x-5)/2)^2+((y-3)/2)^2=\cos^2\pit+\sin^2\pit=1`

for `t=1`, `x=3` and `y=3`. That makes for (`3,3`)
for `t=2`, `x=7` and `y=3`. That makes for (`7,3`)

Based on Wolfram Alpha, the particle moves counterclockwise in a circular manner from (`3,3`) to (`7,3`).

Answer 2

In cartesian: circle with centre `(5,3)`, radius 2

Period `T = 2`.

Explanation:

Set it up for the Pytharorean identity like this:

• `x=5+2\cos\ \pi t implies cos pi t = (x-5)/2`

• `y=3+2\sin\pi t implies sin pi t = (y - 3)/2`

Pytharorean identity

• `sin^2 alpha + cos^2 alpha = 1 implies`

• `((x-5)/2)^2 + ((y - 3)/2)^2 = 1`

`implies (x-5)^2 + (y - 3)^2 = 2^2`

Circular motion with centre `(5,3)`, radius 2

Remember that `cos omega t` or `sin omega t` implies motion with constant angular frequency `omega = 2 pi \ f = (2 pi)/T`.

So you can also describe the periodicity of the motion as `T = 2`.

Finally, the interval: `1\let\le2`, which amounts to half a period.

• `((x(1)),(y(1))) = ((3),(3))`

• `((x(2)),(y(2))) = ((7),(3))`

From the graph, you can see that it's making the CCW journey along the bottom half of the circle in that time interval

graph{ (x-5)^2 + (y - 3)^2 = 2^2 [-10, 10, -5, 5]}