The Math of Driving Way Too Fast.

by Steve Baker.

How fast can a MINI Cooper S go? The specifications for the car say 135mph is the top speed - but can we use math to demonstrate that?

When you are driving at a steady speed in a straight line on a level road, your speed is limited by the amount of engine power it takes to overcome the air resistance and the friction of the wheels against the road. But at high speeds, the air resistance is by far the most important thing.

The Physics formula for air resistance says that the drag on your car is proportional to the square of the speed - and the amount of power consumed to overcome that drag is proportional to the cube of the speed:


    Fd = Cd x 0.5 x rho x A x V-squared

    P = Fd x V

Where:

P = Power (in Watts) required to overcome the air resistance.
Fd = The drag force (in Newtons)
Cd = Coefficient of drag (0.36 for MCS, 0.35 for MC)
rho = Density of the air (about 1.22 kg/cu.m - depending on temperature, pressure, and humitidy)
A = Frontal area in sq.m. (about 2.0 sq.m I estimate).
V = Forward speed in m/sec (Top speed for a MINI Cooper 'S' is 135mph which is 60.35 m/sec).

So at 135mph, the force of the wind on the front of an MCS is 1600 Newtons - about a fifth of a ton - the weight of two good sized baby elephants! The power required to overcome that amount of drag is 96.5 kilowatts - or 129.4 horsepower in quaint american units.

With a 165HP engine (which is 123 kilowatts), about 78% of the power of the engine is needed just to overcome the air resistance. The remaining 35.6 horsepower is required to overcome the 'rolling resistance' of the wheels against the road.

In an aerodynamics web site that I'm rather fond of it says:

"A modern car (drag coefficient 0.34) with a 110 kW (150 HP) engine will be using about 4.6 kW (6 HP) of power to overcome air drag and 11 kW (14 HP) for mechanical propulsion (rolling resistance etc.) when it is being driven at a constant 80 km/h (22.2 m/s, 49 mph). When driving at its top speed of 210 km/h (58 m/s, 128 mph) it will be using 82 kW (112 HP) to overcome drag and the remaining 28 kW (38 HP) to overcome rolling resistance etc." They are saying that their hypothetical car needs 38 HP to overcome the rolling resistance - which pretty close to the 35.6 HP that I estimate that the MCS has available at 135mph.

What all of these numbers show is that the MINI probably can't go faster than 135 unless you can either reduce the drag or increase the horsepower.

In all of this math, the speed we are measuring is the speed through the air - not the speed along the ground. So driving at high speeds into a 10mph headwind will pretty much directly reduce the speed measured on your speedometer by 10mph. At slower speeds, your 'rolling resistance' will be much more important in determining your speed than air resistance - so you don't feel the effect of the wind anywhere near as much as when you are driving fast.