RoseHulman Math/Physics Challenge

Given a point P and a line segment AB, determine the point Q on AB which a bead sliding under the influence of gravity (without friction or any other resistance) on the segment PQ will reach first. 

To put this into a realworld context, imagine that Agent P is traveling by zipline to the AB building to meet Agent Q. Agent P wants to get from his location at the top of building X to the AB building as fast as possible, so Agent P needs to locate the end of his zipline (shoot an arrow) at what point of the AB building and tell Agent Q to meet him there. As a first case, we consider a frictionless zipline and no resistance to Agent P's motion on the zipline, with the motion being purely under the influence of gravity (and no deflection of the line by Agent P's weight). HINTS: For convenience, set point P = (0,p), point A = (a,p) and point B = (b,0). Treat numbers a, b, and p as positive numbers. 
Bonus Problem 1: How does the result change with a frictional force on the zipline with the frictional force opposing the motion and proportional to the component of the weight that is perpendicular to the line PQ? 

Bonus Problem 2: How does the result change with a resistance force proportional to the velocity of Agent P's motion? 

Research Question: How would one account for the deflection of the line under the weight of Agent P? What information would you need to know about the zipline?

REMARKS: The original problem and bonus problem 1 only require geometry, algebra and Newton's laws of motion to solve. Bonus problem 2 requires calculus to set up and solve. The research question requires even more sophisticated mathematics and physics to setup and solve. 

