I am trying to solve for when a circle and a line rotating about a point will collide:

Updated:

I am trying to solve for.T

The arrow fromtoBorepresentsBor the velocity of the circle.Bv

The arrow from the bottom line torepresentsPor the change of the rotating line.θv

The dashed line and circle represents a possible solution to the problem and I only put it there to help me think through the problem.

Constants:

: length of the rotating line (L)CP

: point the line rotates aroundC

,Cx: components ofCyC

: initial point/location of the circleBo

,Box: components ofBoyBo

: velocity of the circleBv

,Bvx: components ofBvyBv

: radius of the circler

: initial rotation of the rotating line (θ)CP

: rotational velocity of the rotating line (θv)CP

Variables:

: end point of the rotating line (P) at timeCPT

,Px: components ofPyP

: normal of the line rotating line (N) at timeCPT

,Nx: components ofNyN

: point/location of the circle at timeBT

,Bx: components ofByB

: distance from the center of the circle (D) to the rotating line (B) at timeCPT

: time (from 0 to 1)T

In the last equationis confusing and a bad choice for a variable. It simply representst+θ*θvas stated.T

Here is my reasoning:

At the top I have the positions of the endpoint of the rotating lineand the center of the circlePdefined as functions of timeB. In order to determine the position of the collision, I must find at what timeTthe distanceTfrom the center of the circleDand the lineBis equal to the radius of the circleCP, orr-D= 0. The distance from a point to a line can be found by taking the dot product of the normal of the line (r) and the point (N). Knowing this I simply expand the equation until all the variables are present and I am left with the final equation at the bottom. I re-condensed the equation into terms ofB,a, andbhoping it would help me solve the problem. If I can gettout of the sine and cosine functions, I would be able to solve the problem.t

I can't figure out what to do from here. I have been staring at this for days.

Thanks,

- Mike