Some falcons, such as peregrines (Falco peregrinus), attack their prey in the air at the end of high-speed dives and are thought to be the fastest of animals. Estimates of their top speed in a dive range up to 157 m s-1, although speeds this high have never been accurately measured. This study investigates the aerodynamic and gravitational forces on 'ideal falcons' and uses a mathematical model to calculate speed and acceleration during diving. Ideal falcons have body masses of 0.5-2.0 kg and morphological and aerodynamic properties based on those measured for real falcons. The top speeds reached during a dive depend on the mass of the bird and the angle and duration of the dive. Given enough time, ideal falcons can reach top speeds of 89-112 m s-1 in a vertical dive, the higher speed for the heaviest bird, when the parasite drag coefficient has a value of 0.18. This value was measured for low-speed flight, and it could plausibly decline to 0.07 at high speeds. Top speeds then would be 138-174 m s-1. An ideal falcon diving at angles between 15 and 90 degrees with a mass of 1 kg reaches 95 % of top speed after travelling approximately 1200 m. The time and altitude loss to reach 95 % of top speed range from 38 s and 322 m at 15 degrees to 16 s and 1140 m at 90 degrees, respectively. During pull out at top speed from a vertical dive, the 1 kg ideal falcon can generate a lift force 18 times its own weight by reducing its wing span, compared with a lift force of 1.7 times its weight at full wing span. The falcon loses 60 m of altitude while pulling out of the dive, and lift and loss of altitude both decrease as the angle of the dive decreases. The 1 kg falcon can slow down in a dive by increasing its parasite drag and the angle of attack of its wings. Both lift and drag increase with angle of attack, but the falcon can cancel the increased lift by holding its wings in a cupped position so that part of the lift is directed laterally. The increased drag of wings producing maximum lift is great enough to decelerate the falcon at -1.5 times the acceleration of gravity at a dive angle of 45 degrees and a speed of 41 m s-1 (0.5 times top speed). Real falcons can control their speeds in a dive by changing their drag and by choosing the length of the dive. They would encounter both advantages and disadvantages by diving at the top speeds of ideal falcons, and whether they achieve those speeds remains to be investigated.