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

[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!

 

[PeroK]

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 $\Delta t$.

 

[PeroK]

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.