For a given power output, what is a rider's speed and distance as a function of time? 

Power, Given Speed and Speed, Given Power describe rider parameters under equilibrium conditions where the power input, speed, and forces acting on a rider are all in balance.  Reaching equilibrium can take over a minute in a time trial under constant power and constant external conditions.  Often we want to know how long it will take a rider to cover a distance from a standing start where equilibrium may never be reached.  In the case of a pursuit, power, speed and forces may not  reach steady state until close to the end of the pursuit.  Differential equations describe the distance a rider covers, speed, and acceleration as functions of time.  This section describes the differential equations and solves them for a riders distance, speed, and acceleration, all as functions of time. 

The sum of the forces acting on a rider is equal the the mass of the rider multiplied by the acceleration of the rider: 

       Sum Forces = m a 
  
The power applied to a rider divided by the speed at which the power is applied gives the force acting on the rider due to power being applied: 

      F_p = P\v 
  
The forces on a rider become: 

      P\v - (A  C_w  Rho v²/2 + Wkg  C_rl + Wkg  Grad) =   Wkg  a 
  
The differential equation can be written as: 

      P\d'[t] - (A Cw  Rho d'[t]²/2 + Wkg  C_rl + Wkg  Grad) \ Wkg =d''[t] 

where for a rider as a function of time d[t] is position, d'[t] is speed, and d''[t] is acceleration.  The solution to this differential equation gives explicit forms for these quantities. 

If results in some parts of the plots are different from what you expected, it may be because time and distance are not consistent, for example, it may take longer to reach a distance than allowed by the time parameter. 

Example plots of equations of motion:

Rider Time at 500 m = 54.3 s.

Rider Speed at 500 m = 11.1 m\s.

Rider Distance vs Time

Rider Speed vs Time

Rider Acceleration vs Time

Rider Speed vs Distance