The balancing speed of the car is reached when the tractive effort at the drive wheels is equal to the combined resistance to motion. So to predict performance we need to understand drag.

We’ll consider three sources of drag:

• rolling resistance
• aerodynamic resistance
• gravity (if there are hills)

Each of these can be a bit iffy for various reasons. We just do the best we can.

rolling resistance

For this project we’ll assume that the resistance to rolling is proportional to the load that a wheel carries. This is a common assumption for cars. There are other approaches, for example when estimating rolling resistance for railway wagons.

Resistance proportional to load is expressed as a coefficient of rolling resistance:

DR = CR*W              DR is rolling drag, CR is coefficient of rolling resistance, W is weight (note, weight is a force measured in Newtons; it is mass in kg times the acceleration in free fall due to gravity (9.8 m/s/s))

Values of CR  found on the web are typically as below:

vehicle	CR
steel wheel on steel rail	0.001 – 0.002
Bicycle tyre on various surfaces	0.001 – 0.004
car tyres on paved roads	0.01 – 0.02
car tyres on solid sand (loose sand higher)	0.04 – 0.08

The wide range of CR can introduce uncertainty into predictions.

Aerodynamic drag

Aerodynamic drag is modelled by a beautiful formula:

DA = 0.5*rho*A*CD*v^2         DA is aerodynamic drag,  rho is the density of air, A is the frontal area, CD is the drag coefficient, v is the velocity of the air over the body and the ‘^’ sign indicates raising velocity to a power – in this case 2, (in other words you square it)

Typical values for CD found on the web are tabulated below.

object	CD
car where effort has been applied to minimise CD	0.2 – 0.3
typical saloon car	0.3 – 0.4
convertible open top	0.6 – 0.7
bus	0.6 – 0.8
bicycle	0.9 – 1.1
truck	1

Once again the range of CD is wide. If it is significant then wind tunnel tests can give more precise values of CD for a specific item.

Gravity

The component of the weight of a vehicle acting down an incline is::

DG = W*sine(theta)             where DG is gravity drag, W = weight (see above)

This is precise, but the uncertainty in this case comes from not really knowing what the incline is.

So the total drag is:  DR + DA + DG

We’ll apply this next time.