Question

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

Answer 1

\[x=5sint~~~y=2cost~~~~~-\pi \le t \le 5\pi\]

Answer 2

@ganeshie8

Answer 3

from the first glance it looks like an ellipse, what do u think ?

Answer 4

I understand it's an ellipse but I don't understand why it moves clockwise around the ellipse 3 times, maybe just the drawing I don't understand haha.

Answer 5

period of sint is 2pi
so one complete revolution = 2pi radians, yes ?

Answer 6

we don't know orientation yet,
but we can see that [-pi, 5pi] has 3 full periods in it : (5pi - -pi)/(2pi) = 6pi/2pi = 3

Answer 7

so it makes 3 full revolutions starting at SOME point and moving in SOME direction

we need to find these starting point and direction

Answer 8

Ah right right, thank you so much :P!

Answer 9

at t = -pi,
(x,y) = (0, -2) :

Answer 10

thats the starting position,
you need to figure out whether the particle moves in clockwise or counterclockwise

Answer 11

we need to plot atleast TWO more points to know the orientation unambiguously

Answer 12

Yeah I got it, as t goes from - pi to 5 pi, (0,-2) is the start point and moves clockwise around 3 times. Thanks Ganeshie ^.^

Answer 13

:) yes and we need to plot the points within the same period.. otherwise there will be a clash

Answer 14

or else if we recall transformations in xy plane :

(x, y) = (cos, sin) : counterclockwise rotation
(x, y) = (sin, cos) : clockwise rotation

Answer 15

that tells immediately the orientation of particle

Answer 16

Yup, can I ask one more just to confirm?

Answer 17

yeah sure il try

Answer 18

\[x=3+2cost~~~y=1+2sint~~~~~~\pi/2 \le t \le 3\pi/2\]

for this what I thought of was x = h+rcost
y = k+sint which gives an equation of a circle

so I got (x-3)^2+(y-1)^2=4

Answer 19

start point (3,3) moving counterclockwise

Answer 20

looks perfect !
but do you really get a full circle ?

Answer 21

No it should be half of a circle I think

Answer 22

you should draw it

Answer 23

y=k+rsint* yeah I'm drawing it now haha

Answer 24

one half of a circle

Answer 25

diagram

Answer 26

diagram tells everything - startpoint, endpoint and orientation