“Be kind, for everyone you meet is fighting a hard battle” - Often attributed to Plato but likely from Ian McLaren (pseudonym of Reverend John Watson)

Sunday, June 24, 2007

Top speed

Those who may have stumbled upon my little blog have probably noticed that I enjoy putting numbers to things. The Land Rover LR3 HSE is certainly an example, though the figures that I've been able to derive have borne only a passing resemblance to measured values. So I think I'll derive one that I won't be able to disprove myself - the maximum speed of the vehicle.

As we've previously noted, in unaccelerated travel the sum of the forces acting on the vehicle must be zero. The rated power of the 4.4 liter V8 engine in the LR3 is 300 horsepower. We'll convert that to 223710 watts using the Google Calculator. Now I'm going to estimate, based on various sites I've visited, that the efficiency of transmitting the engine's power at the flywheel to the road is 78%. This may seem a little low to some, typical figures are often in the 80% to 85% range. But this is a full time four wheel drive vehicle, so losses will be higher.

That leaves 174794 watts to the road. Now, power is force times speed, so if I add the external forces and multiply them by the speed, I'll have the power being utilized to overcome forces. The aerodynamic drag reduces to 0.7491*s^2 and the road load is estimated to be 14.65*s, where s is the speed of the vehicle. If anyone leaves a comment that they would like to know where these figures came from, I'll be happy to oblige.

In any case, since those are forces and speed times force equals power, we'll multiply by speed (s) and equate it to 174794 Thus: 174794=0.7491*s^3+14.65*s^2. I have various computer algebra systems, but the simplest is called Derive 6. Much to my regret, Texas Instruments will stop development and shipment of the program this week. I've had most versions, and where truly exotic math isn't required, I prefer it to Mathcad, Maple, and Mathematica, as capable as those programs are.

In any event, Derive easily solves this cubic equation and determines that s=55.65 meters per second, or 124 miles per hour. For purists, this is the solution in the real domain. I'm not sure how realistic this is - I haven't come anywhere close to topping out the speed in the LR3, nor do I intend to do so. This is the basis of my contention at the outset of this post that I won't be able to disprove the number. However, Car and Driver's site gives the top speed as 121 m.p.h. but refers to it as "governor limited." According to my calculations, you'd have to be going down a hill to exceed this by more than 2.5% so I'm not sure why a governor is needed. I still find this published number satisfying.

1 comment:

Tomislav Veleckovik said...

Thanks for the formula man, I really need it ;)