Question

Describe the motion of the particle with position (x,y) as t varies with the given interval.

x = 5 sin t, y = 2 cos t
-pi<= t <= 5pi

so i'm guessing i first eliminate the parameter and then graph this?

Answer 1

i think i got it.

x = 5 sin t
x/5 = sin t

y = 2 cos t
y/2 = cos t

and since x^2 + y^2 = 1
i can write this as

x^2/25 + y^2/4 = 1
which is in the form of an ellipse

Answer 2

0__________________0 k den i definitely don't know how to do this one xD i think u need someone smarter

Answer 3

@Directrix

Answer 4

starting point would be when t = -pi
so the respective (x,y) values would be:
x = 0 , y = -2
and ending points are the same (0,-2).
all that's left is the motion of the particle which i do not know how to describe.

Answer 5

wait i think i got it
since its from -pi < t < 5pi
sin and cos both have a period of 2pi

so the interval of t is 5pi - (-pi) = 6pi
so the particle would make 6pi/2pi = 3 revolutions
but what about the direction of the particle? how do i determine that?

Answer 6

So you just want the direction of the particle at a point on the curve? It will look something like this (lol):

So that direction vector is really tangent to the curve. That's great news, cause we usually think of tangent vectors as being derivatives AND the derivative should give us the velocity which is literally adding to our position vector to where we're going so it should point where we're going.

Uhhh but we just want the direction not the speed, so we can divide the velocity vector's magnitude out. I can help fill any details in and/or check your work if you wanna go at it.

Answer 7

not at a point on the curve but rather the direction at which the particle is travelling along the ellipse (clockwise/anti-clockwise) how would i determine that exactly?

Answer 8

There are two options here : clockwise, counterclockwise

Answer 9

yeah counter*clockwise