[To understand the material of this document, I assume the readers have a firm understanding of previous pages, especially the 0-100km/h page. Familiarity with integration is also desirable.]
In addition to the 0-100km/h time, car acceleration is often also measured in 0-400m time. (or quartermile time in the UK/US) Some may ask: why 400m? I think it has to do with the drag racing culture. Most race tracks have a long straight line near the finish line, usually much longer than 400m such that cars have enough distance to come to a full stop. To cater to the drag racing crowd, they let the drag racers to run 400m with their own cars and give them a time slip at the end of each run. The drag racers can then use the time slips to compare how well they run their cars and/or how well they modded their car.
In this page, we are going to explain the math that would allow us to estimate the 0-400m time. It will use a lot of materials from the other pages. So please be give yourself a quick refresher of the previous materials before you proceed. You have been warned!
Let's recall the formulas for dv in the 0-100km/h page. By using the results from the Optimal Shift Point page. We can obtain the dv formulas for every gear.
Using the same integration technique in the 0-100km/h page. We obtain indefinite integral formulas for t.
Here we see that in the 5th gear, the funciton is not defined for v > 71.5m/s. If we plot out the function, we will see that the speed approaches 71.5m/s (257.4km/h) asymptotically - meaning that the car can never accelerate beyond that point. This is the maximum speed that the car can attain due to air resistance. This value is consistent with the result in the Max Speed page, although slightly lower here as we include rolling friction.
By plugging in the boundary conditions, we can obtain the contant terms for the indefinite integral formulas for t.
Here we can obtain the 0-100km/h time (ie 0-27.8m/s time) by
Now we can proceed to find out the 0-400m time. To do this, we need to obtain the formulas for displacement d. The first step is to find the formulas for v by inverting the formulas for t.
By integrating v with respect to t, we can have the indefinite integral formulas for d.
By plugging in the boundary conditions, we can obtain the constant terms for the indefinite integral formulas for d.
Using these formulas for t, v and d, we can obtain the following table.
This shows that the car reaches 400m in the 4th gear. Now we can solve for the time to go from 0-400m.
This result is again a bit higher than the 12.978s quarter mile time the car magazines got. You can refer to the bottom of the 0-100km/h page for some possible explanations for the difference.
We have plotted out the distance (red line) and velocity (green line) versus time. This allows us to simply read out any 0-X km/h time and 0-X m time.
Wow, congrats for you making it through all the tedious math to get here! But isn't it rewarding to see that all the math in previous pages all come together in a consistent and elegant way?
Or if you are a hardcore drag racer, isn't it nice to test run your imaginary modded car with the math presented here before you pluck down your hard earned money on the mods?
First Draft: March 13th, 2009 |
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