# Tesla Model S

The Tesla Model S can go from 0 to 60 mph (96 km/h) in 2.5 seconds, in what the company calls ”Ludicrous mode,”, the company said in a statement. The only commercial cars on the planet that can beat the Tesla Model S, the LaFerrari and the Porsche 918 Spyder, each cost about \$1 million and are ”tiny” two-seater roadsters. Is there a limit how fast it is possible to accelerate?

Traction is the possibility for a car to accelerate. As a result of the driver pushing the throttle, an engine revs up and puts down all torque onto the wheels in order to speed up. tyre works best under a very slight wheel spin. If there’s too much wheel spin, the tyres lose traction and acceleration is greatly reduced.

Of course you need anti-spin control, but this is a clever way to figure out what the limit is. What is the maximum braking power with an ABS? See this video for the answer.

## An engineers perspective of an electric car

Let us assume that you have the right to use a Tesla Model S to travel a very long distance. What is the optimal velocity if the sequence is charging, driving repeated for ever? Further, let us assume that only the drag is considered, e.g. the rolling resistance is ignored. Then the optimization problem could be described as:

$\min_v 0.5, \rho c_{dA}, v^2 \frac{1}{P}+\frac{1}{v}$

where v is the velocity, c_dA the drag coefficient, P the power.

The ”performance parameter” is the ratio of the charging power and the drag coefficient.

## An economist perspective  electric car

Let assume that you want to find the optimal velocity regarding energy consumption vs traveling time.  According to ”Trafikverket” one hour is worth 108 SEK at year 2010. Let us further assume only drag is considered as load of the vehicle. Further, the density is calculated as  at sea altitude International Standard Atmosphere with the cost of electricity 1 SEK/KWh, for Tesla Model S and Volvo XC90 2015 Diesel with 35 % efficiency in the internal combustion engine. The price for the Diesel at Göteborg and US are 12.32 and 5.40 SEK/litre respectively. The difference is in the CdA parameter (0.576 and 0.92 for Model S and XC 90).

The optimum velocity is very different in these circumstances. In the case of a electric price of 5.5 SEK/kWh the price curve of the Model S and the XC 90 in Göteborg are the same.

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