Parametrics - intersection and collision

[Pi Face]

Homework Statement

x1t= 2-cos(pi*t)
y1t= 3+7sin(pi*t)

x2t= 3t+2
y2t= -(7/15)(3t+1)² + 157/15

Find points of intersection and collision

Homework Equations

above?

The Attempt at a Solution

Well, to find the intersection I think I need to eliminate the parameters for both equations so I get an equations in terms of x and y, and then set them eqaul to each other.

x=2-3cos(pit)
3cos(pit)=2-x
cos(pit)=(2-x)/3

y=3+7sin(pit)
7sin(pit)=y-3
sin(pit)=(y-3)/7

using the rule cos^2+sin^2=1 and substituting, we get

((y-3)/7)^2 + ((2-x)/3)^2 = 1

I guess we should solve for y here?

Seems a little unwieldy to simplify; is there a trick or something?

Anyways, I used the solve function on my calc and got

y= (7*sqrt(-x^2 + 4x+5) +9) / 3


now on to the other curve

x=3t+2
3t=x-2
t=(x-2)/3

plug the equations for t into the y equation

I really don't know how to deal with this manually. the y2t function is complicated enough by itself, and now i have to replace t with a fraction.

I did however graph the function and i think maybe the answers are t=1 which is (5,3) and t=1/2 which is (2, 10)
not entirely sure.

and then for collision, we set the x functions to be the same?

2-3cos(pit)=3t+2
3cos(pit)=3t
cos(pit)=t
I used a calc and got t= -1, -0.7898, 0.377
can the last two values be represented in fractional from with pi?

now i do the same thing w/ y functions

3+7sin(pit)= -(7/15)(3t+1)² + 157/15

do i have to pretty much use a calculator for this?

anyways, i got t=-1.208, 0.307, 1, 1.52

none of the t values are the same.

in desperate need of help please >.>

 

[Pi Face]

anyone? =(

even partial help would be appreciated

 

[Dick]

Ok, I'll give you partial help. You probably meant "x1t= 2-3*cos(pi*t)" in your problem statement, right? Since that's what you used in trying to solve it. I think you are taking the correct approach. (2,10) is an intersection. I don't think it's a collision, can you tell me why? There's another real intersection as well, which I'm pretty sure is not a collision. You are pretty good at this, just keep working on it and I'll check in tomorrow. Sorry, it's really late here.