# Transition time from one steady-state Speed to another for a cyclist

• cyclemark
In summary, the speed estimator calculates a cyclist's steady state speed v given their power output p (watts), rider mass m (kg) and hill gradient g. For this particular model, headwind and tailwind beyond the drag force conditions are not relevant. The estimator uses three forces against the cyclist: gravity, rolling resistance and drag. The estimator finds the cyclist's steady state speed by solving a differential equation that takes into account m, c, cdA, rho, g, and v. The time it takes to reach a steady state speed is estimated by taking the initial acceleration and dividing it by the incremental change in speed.
cyclemark
I have a speed estimator that calculates a cyclist's Steady State Velocity v given the rider's power output p (watts), rider mass m (kg) and hill gradient g. For this particular model, headwind and tailwind beyond the drag force conditions are not relevant.

In the model I use, there are three forces against the cyclist:
• Gravity: function of m and g,
• Rolling Resistance: function of m and g and coefficient of rolling resistance c
• Drag: function of v^2, CdA and air density rho.
Then I add them up to get to the total resistive force F_total.
So F_total is a function of m, c, cdA, rho, g, v.
But, in my situation, m, c, cdA, rho can safely be assumed to be constant, so F_total is really only a function of g and v
And the steady-state speed is reached when p = F_resist ⋅ v.
I currently solve this equation using a mid-point search in a loop feeding in potential velocities until the equation is satisfied with an error of less than 0.0001. But if there is a formula to answer this please let me know!

Now, the tricky part for me, I'm looking for a way to estimate how long it takes to transition from one steady state velocity to another when either p or g or both change.

For example:
if p=300w, g=0% then v=39kph and if p=300w, g=5% then v=20.75kph. But how long does it take to go from 39 to 20.75kph?
Similarly, p=300w, g=0% then v=39kph and if p=400w, g=0% then v=43.35kph. But how long does it take for the speed to increase to 43.35kph?

I hope this explanation is satisfactory. Please let me know if any clarification is necessary.
Any help would be greatly appreciated!

cyclemark said:
Summary:: I know how to calculate a cyclist's steady-state speed given certain parameters. What I don't know how to do is estimate how long it will take to transition from one steady-state speed to another.

I have a speed estimator that calculates a cyclist's Steady State Velocity v given the rider's power output p (watts), rider mass m (kg) and hill gradient g. For this particular model, headwind and tailwind beyond the drag force conditions are not relevant.

In the model I use, there are three forces against the cyclist:
• Gravity: function of m and g,
• Rolling Resistance: function of m and g and coefficient of rolling resistance c
• Drag: function of v^2, CdA and air density rho.
Then I add them up to get to the total resistive force F_total.
So F_total is a function of m, c, cdA, rho, g, v.
But, in my situation, m, c, cdA, rho can safely be assumed to be constant, so F_total is really only a function of g and v
And the steady-state speed is reached when p = F_resist ⋅ v.
I currently solve this equation using a mid-point search in a loop feeding in potential velocities until the equation is satisfied with an error of less than 0.0001. But if there is a formula to answer this please let me know!

Now, the tricky part for me, I'm looking for a way to estimate how long it takes to transition from one steady state velocity to another when either p or g or both change.

For example:
if p=300w, g=0% then v=39kph and if p=300w, g=5% then v=20.75kph. But how long does it take to go from 39 to 20.75kph?
Similarly, p=300w, g=0% then v=39kph and if p=400w, g=0% then v=43.35kph. But how long does it take for the speed to increase to 43.35kph?

I hope this explanation is satisfactory. Please let me know if any clarification is necessary.
Any help would be greatly appreciated!

The quick answer is that the time is infinite, as these speeds are terminal velocities and theoretically never reached! The real question is how close do you want to get to the terminal velocity. Within ##1 kmph##, say.

To get the time, you either solve a differential equation or do a nhumerical estimate. E.g. Take the initial acceleration; assume this applies for a small time ##\Delta t##, then recalculate the new acceleration etc. The smaller you take ##\Delta t## the more accurate the estimate. With this estimate, you should be able to hit the terminal velocity in a finite number of intervals of ##\Delta t##.

cyclemark said:
Similarly, p=300w, g=0% then v=39kph and if p=400w, g=0% then v=43.35kph. But how long does it take for the speed to increase to 43.35kph?

I knocked up a quick spreadsheet. I took increments of ##1s##. What I got was:

After 10 seconds: 41.3 km/h
After 20 seconds: 42.4 km/h
After 30 seconds: 42.9 km/h
After 40 seconds: 43.1 km/h

And then, obviously, slow progress after that.

## 1. What is "transition time" for a cyclist?

Transition time refers to the time it takes for a cyclist to change their speed from one steady-state to another. This can include accelerating from a stop, or changing gears to increase or decrease speed.

## 2. Why is transition time important for cyclists?

Transition time is important for cyclists because it can affect their overall performance and efficiency. The longer the transition time, the more energy and effort the cyclist will need to expend, which can impact their endurance and speed.

## 3. What factors can affect transition time for a cyclist?

Some factors that can affect transition time for a cyclist include the terrain (e.g. uphill or downhill), wind resistance, the weight of the bike and rider, and the cyclist's physical fitness and technique.

## 4. How can a cyclist improve their transition time?

A cyclist can improve their transition time by practicing and improving their technique, such as learning how to shift gears efficiently and smoothly. They can also work on their overall physical fitness and strength, which can help them accelerate and change speeds more quickly.

## 5. Is there an ideal transition time for cyclists?

There is no specific ideal transition time for cyclists, as it can vary depending on the individual's abilities and the specific situation. However, generally, a shorter transition time is preferred as it can help conserve energy and improve overall performance.

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