Curve for a ramp resulting in shortest time possible?

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To determine the optimal ramp shape for an object sliding from (xi, yi) to (xf, yf) in the shortest time, the discussion highlights the need to consider the principles of mechanics and acceleration due to gravity. The object accelerates down the ramp, and while a straight drop maximizes velocity, it may not be the most efficient path. The conversation references the Tautochrone and Brachistochrone curves, which are relevant to this problem but may exceed the current curriculum's scope. The participant expresses uncertainty about applying calculus of variations, indicating a focus on basic mechanics instead. Understanding the balance between distance and acceleration is crucial for solving this problem effectively.
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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.
 
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This normally involves calculus of variations, are you supposed to know how to do this?
 
No prior knowledge of calculus of variations. We're just basing this off of mechanics.
 
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.
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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