Following a survey of existing mathematical models of running, a new analysis is developed, based on the first law of thermodynamics. The method properly accounts for each term in the energy balance, and avoids the use of mechanical efficiency factors. A relationship is derived between race distance and the time taken to run that distance. An excellent correlation of results from recent Olympic Games is established for events over distances from 100 m to 10,000 m. The velocity-time relationship for a sprinter running 100 m at maximum available power is obtained by numerical integration of the power equation. It is shown that the peak velocity is achieved in the middle stages of the race, a result which is consistent with practice, but which previous calculations based on Newton's laws have failed to predict. Further applications of the analysis are indicated.