The other big problem with any linear calculation is that the thrust available from the electric motors is not constant, and not even linear.Fabulist said:
Consider the torque graph here, of a typical electric-pump motor: http://www.engineeringtoolbox.com/electrical-motors-torques-d_651.html
Horsepower is torque x RPM. In an ideal motor horsepower remains constant as torque approaches infinity (because torque at 0 is infinity, and torque at infinity is zero, and in between, the product is a constant, so the limit from above zero and the limit from below infinity are the same constant as the horsepower anywhere in between). No electric motor is ideal, though. Here's a graph of an actual EV electric motor:
http://www.evme.com.au/performance/power/
The green line is horsepower (well, actually kW output as read off the right hand scale, just multiply by 4/3 to get hp). The red line is torque. The maximum available torque (about 225 Nm) is essentially constant, rather than rising to infinity, below about 3500 RPM.
I don't have a torque curve for the Karma's electric motor, but we do know the maximum value (1300 Nm). We can find an approximation though ... let's assume it's flat up to about X mph and then drops in a nice curve from then on, giving linear horsepower (which we know is ~403 hp, which is 300 kW) as in an ideal electric motor.
Now, actual power output (in watts) is torque (in newton-meters) times 2pi times rotational speed (in revs/sec aka RPS). (see: http://en.wikipedia.org/wiki/Torque)
The Karma's rear tires are Goodyear 285/35R22 with an outside diameter of 29.9 inches (0.759 meters). To go 1 mile, this tire must turn (1 mile / (29.9 pi) inches) times, or about 880.7 times. At 60 mph, one mile takes 60 seconds, so the tire turns 880.7 rotations per 60 seconds or about 14.7 RPS. We've assumed peak horsepower arrives at X mph (and remains constant from then on), so we get our peak output at X rather than 60, or X/60ths of this RPS. But hang on a second, that RPS is tire rotation speed, not motor speed. We need to put in the correction factor for the final drive ratio, which is 4:1.
Plugging all these in, we have: 300,000 = 1300 * 2pi * (X/60) * 14.7 * 4. Now we just need to solve for X. Move all the constants to the left, giving: 60 * (300000 / 1300 / 2pi / 14.7 / 4) = X, and compute. The result I get is about 37.5.
If this is correct, then the power output of the electric motors rises linearly with speed up to 37.5 mph. (This is because power is torque—which we've assumed is constant up to X, which we just calculated as 37.5 mph—times 2pi times RPS, and RPS is linear with speed.) After that the power has hit the 300 kW limit and remains constant.
(This fits pretty well with the feel of the car, where the acceleration eases up a bit above 40 mph.)
This all means that 0-60 time is not determined by 300 kW, but rather by the integration of the variable power output up to 37.5 mph, followed by the fixed 300 kW output from 37.5 to 60. I will leave that calculation for someone else, or another day, because it is almost 3 AM here now...
