The Car Equation

The physics of automotive motion is complicated, but the basics are clear from a relatively simple equation. It’s worth posting, as it illustrates some of the main challenges faced by design engineers and drivers who aim to increase fuel economy. 

So, ignoring secondary effects, here is what might be termed the fundamental equation of automotive force: 

F = mgC_rr + ½ρC_DAv² + ma + mgsin(θ)

where 

F = force required at the wheels of the vehicle
m = mass of the vehicle
C_rr = coefficient of rolling resistance between tires and road surface
ρ = density of the ambient air
C_D = coefficient of drag of the vehicle in the direction of travel
A = cross-sectional area of the vehicle
v = speed in the direction of travel
a = acceleration of the vehicle
g = local acceleration of gravity
θ = angle (relative to horizontal) of the road surface

To increase fuel economy, engineers work to increase the efficiency of the drive train that delivers the force that’s required at the wheels. This is complicated work, and they have been at it for decades. But only recently has attention focused on the capture, storage, and re-use of kinetic energy that is normally lost when vehicles slow down. Electric hybrids are just the beginning. 

On the other side of the equation, engineers work to decrease the amount of force that’s required. From that side, you can see why they try to do three things based on variables that are primarily under their control (vs. the driver’s control): 

• reduce the vehicle mass (particularly important, since there are three terms in the force equation that are proportional to the mass, and because vehicles today are made from heavy materials) 
• reduce the rolling resistance (tires) 
• improve the aerodynamics (reduce cross section, reduce coefficient of drag)

Of course, there are economic and other tradeoffs involved. Reduce mass, but don’t give up too much safety; reduce rolling resistance, but don’t give up too much wear or grip; improve aerodynamics, but don’t give up too much comfort, convenience, and carrying capacity (and don’t make it ugly!). 

While these variables are primarily under the design engineer’s control, the driver has some control as well. If you load up the car, the mass increases. If you put stuff on the roof, the aerodynamic drag increases (likewise if you open all the windows). If you choose the wrong replacement tires or don’t keep the tires properly inflated, the rolling resistance increases. 

On the other hand, the equation shows the importance of variables that are primarily under the driver’s control: 

• acceleration - how often and how strongly you accelerate 
• speed - particularly important given that the second term is proportional to the square of your speed (drive a little faster, and there’s considerably more drag) 
• road surface – where you drive affects rolling resistance 
• hills - how often you go up them and how steep they are; (the sign of the last term becomes negative when you go back down, but you never get back all you lose going up. Although there are many variables involved - speed, cornering, braking, regeneration, etc. – at best you can recover most of the potential energy gained by climbing the hill. However, the energy spent going uphill is considerably greater than that potential energy gain since engines and drivetrains are not 100% efficient. The difference is lost forever.)

As mentioned, these variables are primarily under the driver’s control; but they are also affected by vehicle design. For example, drive trains may be most efficient at a particular speed (so driving very slowly may not yield the best fuel economy). And, obviously, you can’t accelerate or drive faster than the vehicle’s capabilities (which may be electronically limited). Similarly, unless you’re an off-roader, your choices of road surface and terrain are limited strongly by those in charge of road construction and maintenance.

Finally, note that there’s an important effect that’s beyond the scope of the equation above, namely that fuel economy is reduced by diverting energy for purposes other than forward motion – in particular for comfort and convenience features, including climate control, on-board electronics, etc. Here, the engineer is responsible for how energy-efficiency the features are, while the driver is responsible for how much they are used.

Note: post revised on 12 November, 2007; see comments; js