# Downhill to uphill roller coaster velocities on a bike

• ErikMM
In summary, the conversation revolves around the problem of determining the equations needed to calculate speeds on a mountain bike going downhill and uphill on dirt hills of varying grades and lengths. The conversation also mentions the consideration of factors such as friction, drag, suspension, and rider and bike mass. The ultimate goal is to design a scenario with sin-wave-like rollers, jumps, and clothoidal jumps to maximize trail length. The conversation also mentions specific assumptions and asks for suggestions on how to approach the problem.
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?

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?

## 1. How does a bike maintain its velocity going downhill on a roller coaster?

A bike maintains its velocity going downhill on a roller coaster due to the principle of conservation of energy. As the bike moves down the hill, it gains potential energy, which is then converted into kinetic energy as it moves up the next hill.

## 2. Can a bike reach higher velocities going uphill on a roller coaster?

Yes, a bike can reach higher velocities going uphill on a roller coaster. This is because as the bike moves up the hill, its potential energy is converted into kinetic energy, resulting in an increase in velocity.

## 3. Is it more difficult to pedal a bike going uphill on a roller coaster compared to going downhill?

Yes, it is more difficult to pedal a bike going uphill on a roller coaster compared to going downhill. This is because the bike has to work against gravity to climb the hill, requiring more energy and effort from the rider.

## 4. How does the weight of the rider affect the velocity of a bike on a roller coaster?

The weight of the rider does not significantly affect the velocity of a bike on a roller coaster. The velocity is primarily determined by the design and layout of the roller coaster, as well as the rider's pedaling power and aerodynamic positioning.

## 5. Are there any safety concerns when riding a bike on a downhill to uphill roller coaster?

Yes, there can be safety concerns when riding a bike on a downhill to uphill roller coaster. The rider should always wear appropriate protective gear, follow all safety instructions, and be aware of potential hazards such as sudden drops, sharp turns, and other riders on the track.

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