Downhill to uphill roller coaster velocities on a bike

[ErikMM]

first off: hello, I'm new here.

Homework Statement

This is more of a personal quest than homework, a problem that I made into homework for myself. I set out to determine the equation(s) needed to determine speeds on a mountain bike at the bottom of dirt hills of varying grades and lengths. Turns out its not so straightforward as I thought being that friction and drag come into play, and perhaps suspension.

Besides the velocity at the bottom of the hill (bdhv=bottom downhill velocity), I then want to determine how far up various uphill angles a rider will be able to coast before coming to a complete stop(uhvz=uphill to zero velocity)...only to go back down again-- a roller coaster. I am guessing this second equation is practically the same as the first equation, only now the initial and final velocities are reversed.

The ultimate goal, a scenario: design sin-wave-like rollers and rollers that can be jumped, as well as clothoidal jumps to maximize trail length given a start and finish elevation point.

Assume:

• the initial speed at the top (or bottom) of the hill needs to be included. It could be 0, or some velocity.
• no pedaling (all coasting), no braking, no sliding down (or up) the hills
• grades are a % greater than zero (there are no flat spots for a measurable distance)
• there are no turns
• factor in downhill and uphill run lengths
• there is rolling resistance or kinetic friction
• there is drag (may need height and weight data to guesstimate the Drag Area)
• bike mass is included (I was under the impression that rider+bike mass cancels, as acceleration for wheels is a=1/2 g sinθ, but the link below includes mass...it seems only for purposes of drag though?)
• elevation above sea level is factored in for air density
• head or tail wind speed factored in if present
• Assume rider won't shift their body much (stand-sit-stand-sit) to affect drag
• Also assume the distance of change in grade, or reversal transition, at troughs and peaks are about 100 inch (2.54m) long sine-wave-like smooth curves. So about 50 inches of transition to reach 0 degrees before beginning to transition up or down again. I imagine this is a trivial component.
• Assume riders don't pump these curved sections for a boost

I think all I really need are equations for bdhv, and upvz described above. Then I can plug in the grades, and rider+bike info as needed. And the scenario can be approached one reversal or roller at a time.

Homework Equations

it looks like most of the equations I need are at the bottom of this page: http://www.kreuzotter.de/english/espeed.htm#forml
I think I know the order in which to solve them, but don't know that they are what I need.

my assumption: solve for a and b (using the equations in the link above), then verify if: a² + b³ ≥ 0 or a² + b³ < 0. Then solve for the appropriate v.

The Attempt at a Solution

That's why I am here.

a. Do you think my assumption above in 2. is right?
b. Will it give me the velocity down and up that I seek?
c. The "boost" or pump is an interesting problem, as most riders will attempt to do this (and even pedal) at the bottom and top of the reversals. Despite pedaling, how could this boost/pump or recoil be factored in?

 

[Nate810]

d. Would it require a completely different equation? e. Is there a way to factor in the mass of the rider and his/her pumping motion? f. Is this problem even solvable, or are there too many variables and unknowns?g. Are there any other equations, factors, or data I am missing? h. Are there any suggestions for a starting point?