"Power to Weight" for Soapbox Racers

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One of the key indicators of a car's performance is its power to weight ratio. The better ratio is, the faster the car will accelerate.

Obviously this can't work for a gravity powered soapbox racer as it has no engine, but there is an equivalent figure.

A cartie's "power" comes from gravity acting on its mass, and the more mass it has the more force there is propelling it down the hill.

Acting against that are rolling resistance, wheel inertia and aerodynamic drag. Drag is by far the largest of these, and when you are going more than about 40mph it accounts for ~75% of your total losses.

At terminal velocity, there is no wheel inertia as the cartie is not accelerating. The effects of rolling resistance can be ignored as they will be negligible and in any case will be broadly comparable for most carties.

This gives us an equation for terminal velocity as follows (see here for all the maths);

V =  √ (2(m g sin(θ) / ρCdA)

i.e. on any given gradient, V is constant for any value of m/CdA

So the soapbox racer's equivalent of power to weight ratio is the mass to drag area ratio; mass / CdA (or M:CdA for short).

Soapbox Mass to Drag Area Ratio In the UK, soapbox racers typically run at around 600 Kg/m2. A slow cartie might only have about 300 Kg/m2, and a very fast one could conceivably reach in excess of 1000 Kg/m2. Your typical car, however, with a CdA of around 0.65m2 and a mass of about 1200Kg, has an M:CdA of about 1850Kg/m2. This explains why your car will roll down a hill very much faster than your cartie.

The graph on the right (click to enlarge) shows the speed profile of three carties with M:CdA figures of 300, 600 and 1000 Kg/m2 respectively. You can see over the first 100m (up to about 30mph) there is very little difference between them. Even after 1Km, there is only 10 seconds between the fastest and the slowest. Soon after that, however, the 300Kg/m2 cartie reaches its top speed whereas the others are still accelerating. The 1000Kg/m2 cartie reaches a top speed of 85.4mph just after the 4Km mark.

scottishcarties's picture

terminal velocity and air density

Another interesting thing to notice about the graph is that the 300 and 600 Kg/m2 profiles (green and blue) show a very slight gradual reduction in speed after they reach their maximum speed. This is because the newest version of CartieSim (not on general release yet) models the increase in air density as the altitude decreases, which will slow the cartie down very slightly.
scottishcarties | April 17, 2014 - 15:31

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