I have to say there are many details about the Model 3 that haven't been fully revealed and this thread really provided the first evidence of that with the EPA certs. I'd like to provide another example, less paper-trail to follow, but uses fairly sound physics.
I opened a thread asking for folks to estimate 1/4 mile times of the LR because I wanted to see if the LR would be competitive with the BMW 340i for that measure (turns out, probably not). I didn't get a satisfactory answer, so I had to not be lazy. Using first order physics I figured if I could reasonably predict the Model S for both 0-60 and 1/4 mile, then I'd have a good enough model to predict the Model 3's 1/4 mile time and since I already know it's 0-60 time I would be able to tweak any differences between MS and M3....
Ok, if you don't want to know the details of the equations, skip down a few lines
- Using Newton's law, F=ma ; a = F/m
- F used to accelerate is force of motor MINUS resistive forces
- Resistive forces, Fresis = Frr + Fdrag where:
- Frr = rolling resistance
- Frr = Crr x v
- Crr = Coeff of rolling resistance (often expressed as ~ 30 x Cdrag)
- v = velocity
- Fdrag = aero resistance
- 0.5 x Cd x A x p x v^2
- Cd = 0.23 (Coeff of drag)
- A = 2.23 m^2 (frontal area, pulled this from some hypothetical threads)
- p = 1.29 kg/m^3 (rho, air density)
- v = velocity
- Force of motor
- Lots of torque and power curves for Teslas floating around with numbered axes. I dropped a scale on one and created a table. For the 3 I scaled the values according to the information available (258 HP from EPA document = 192kW). I then created an M3 motor curve that looked just like the MS, but was 72% of the peak values. With such a flat curve, I think this stands to reason, but will likely be a point of debate
- Fmotor = torque * neff * rfinal / rwheel
- torque is dependent on RPM and in Nm
- neff = transmission efficiency. I used 0.95
- rfinal = final drive ratio (I used 9 from EPA report)
- rwheel = rear wheel (m) I used 235/40R19 to calculate radius for M3 (.3353 m)
- Now for a given change in velocity (ie at 10MPH vs 5 MPH), I can look up the motor torque and calculate both the acceleration and delta t (time in seconds). I can also calculate delta d (distance)
- So in 1 MPH increments, I'm able to calculate accelerations and distance traveled which gives me what I want
So when I plugged in all the values necessary for the Model S (different tires, mass, frontal area, cd, motor curve, final drive, etc), I got what I would consider pretty damn close:
- 0-60 = 4.45 sec
- 1/4 mile = 13.1 sec @ 107 MPH
Not shabby IMO. Great, so let's try the M3! This is where it got interesting......
When I used all the M3 data (tire size, mass, etc.) it came up short from 0-60. It was predicting 0-60 in 5.84 seconds. We know the LR (which is what all the parameters are based off) should be at 5.1 seconds, so what are my levers? Even driving the transmission efficiency to 100% (impossible), only gets me to 5.5 seconds. Lower resistance tires can't get me there. I can't change mass, tire size, final drive ratio, frontal area, or Cd because we actually know these things. I'm open for suggestions of what to try, but my only option was to go to the torque/power curve. Again, the curve is completely flat for basically ALL of the 0-60, so you can't really say it's how I modeled it. I found that if I applied a 1.14x multiplier to the output curve I came in at 5.1 seconds. That would mean that instead of 192 kW motor, it's more like 219 kW (346 HP)! By the way with that 1.14x multiplier I now get:
- 0-60 = 5.1 sec
- 1/4 mile = 13.9 sec @ 101 MPH
Summary:
- I know my first order estimates do NOT consider traction, I'm ok with that for now.
- Something is off with the data, they still aren't coming clean with this car. Either more powerful motor than advertised, lower mass, or ???
- This message is to further the thread...."Tesla has created a monster!" and I don't think they are ready to fess up yet!
- If your interested in the calcs, here's the google sheets:
- Tesla quarter mile calcs.xlsx