I was interested in approximating how much adding a X lb/Y hp electric assist system would shave off my 0-whatever times. The problem is, most 0-60, or 0-anything really, calculators I've found online are wayyyy off. Then I remembered that someone had formalized
acceleration already, so all I had to do was plug and chug. That said, it's a bit jumpy and there's at least one typo, so I figured that messing w/ latex never hurt, and pumped out some intermediate steps.
We start out w/ the tried and true,
Scourge of introductory physics students everywhere.
Then, since we're looking for acceleration in terms of speed, we divide power by speed to get force.
We also know acceleration is just the instantaneous change in velocity over the instantaneous change in time, so w/ some substitution we get...
And a bit more algebra gets us.
Integrate both sides and we get time as a function of velocity.
Which is...
In terms of rolling/aero drag. With tau*v being the power generated by the engine assuming the torque curve is flat. Finding tau is easy enough, just divide peak power output by the maximum speed in whatever gear we're in. We can even get a power curve for a powertrain that doesn't have a flat torque curve, although that's an exercise for the reader.
Now, the problem is, that looks pretty nasty, so we simplify the coefficients...
Pop it into Wolfram's online integrator, and...
BAM!
Still looks kinda nasty, eh? The arctan of imaginary numbers? Bleh. That's O.K. We're lazy, and we know it, and we also know about this nifty free little math program called octave, available for *nix and windows. We grab that, and can now evaluate this little SOB easy peasy. The only thing easier would be if I made a program w/ this algorithm in octave, which I hope I'll get around to later, but I figure this is good enough for a start.
Now we have an accurate approximation of acceleration in whatever gear between whatever speeds, which is great for those interested in dropping in an electric motor or hybridizing who want to get an idea about performance, as well as anyone who wants to see what changes in HP/weight will do for acceleration. As a double bonus, accelerating for a while in whatever gear at higher speeds may also allows us to calculate the drag area with greater resolution, and for those of us who don't know what our drag area is, the manufacturer's 0-60 times, along with some educated guesses, can result in a decent approximation of it. Also, if torque isn't flat resulting in a non-linear power curve, tossing in a polynomial for engine power doesn't change the outcome much, so we can deal with all sorts of drivetrain combos, even a shot of nawzzz, brah.
P.S. It seems pretty accurate too. I plugged in the numbers for my mercedes project, assumin Crr=.1, drivetrain efficiency is 85%, given the different speeds in gear and curb weight, and in order to get a 0-60 time of 27.2s, plus the time needed for 3 shifts, which is supposedly boardline line for .6s shifts, for the stock 29s 0-60, I had to go with a CdA=.75m^2, which seems about right since it's has a little less reference area than the Camryaro, and almost certainly a higher drag coefficient based on looks alone, so overall the drag area should be a bit higher. Fooling around w/ it also shows how halving the drag area can substantially reduce the time needed to accelerate at higher speeds. For example cutting the drag coefficient of the average small pickup truck in half would shave nearly 10s off of the truck's 0-80mph time, and 5s off of it's 60-80mph time.
Edit- physicsforum took out my bandwidth stealing butt, so I'll host the latex images someplace else later.