Curve for a ramp resulting in shortest time possible?

[babblahbah]

Homework Statement

An object slides without friction down a ramp, from (xi, yi) to (xf, yf). What is the equation for the shape of the ramp connecting those two points which would enable the object to reach (xf, yf) in the shortest possible time? Also, describe the shape of the ramp.

Homework Equations

All I've done is set up a free body diagram and the forces acting in the x and y direction. I know that the object is accelerating with a = G*sin(θ).

The Attempt at a Solution

I was thinking something along the lines of having the object drop straight down in order to maximize its velocity and then right before it reaches the same level of the final position, a curve appears in the ramp that launches it in the x direction toward it. Though this would give the fastest final velocity, I'm not sure if it would be the most efficient way. I know the shortest distance would be a straight line connecting the two points but I'm not sure if that helps.

 

[rcgldr]

This normally involves calculus of variations, are you supposed to know how to do this?

 

[babblahbah]

No prior knowledge of calculus of variations. We're just basing this off of mechanics.

 

[babblahbah]

I've been looking into the different types of curves such as the Tautochrone curve and the Brachistochrone curve but everything seems to go beyond what we are currently learning in our class.