0-100 km/h Time

[To understand the material of this document, I assume the readers have a firm understanding of previous pages. Familiarity with integration is also desirable.]

Introduction

I was first introduced to 0-100km/h time (or 0-60mph time for UK/US people) from car magazines. Car magazines use this metric to determine how fast can a car accelerate from 0km/h to 100km/h. Because of that, when people talk about how fast a car can go, the first thing come to their minds is the acceleration time from 0 to 100km/h. Since this is an important indicator of a car's performance, I would like to spend this page to demonstrate a way to calculate this time based on the car data we collected.

In this section, we try to calculate the 0-100km/h time from a theoretical point of view, using the simplified torque curve and results of optimal shift points from previous pages.

Assumptions

If you take a look at the torque curve, you will notice that at 0rpm, the torque is zero, hence there is no power output. In other words, starting at rest, the car cannot accelerate and it will sit still forever.

Obviously this is not the case when you are driving. When you start up a car, the battery in your car will start the engine and run it at a certain speed. This engine speed is called the idle engine speed. When you engage the first gear (or set it to drive in automatic cars), the clutch is engaged at this idle engine speed.

When the clutch engages, the wheels are driven by the torque generated at the idle engine speed. In reality, this phenomenon is similar to clutch dump - rev your engine to a certain engine speed (usually at a high speed) and then engage the clutch. The exact physical effect of clutch dump is still under research by us. For now in order to simplify the calculation, we assume that when clutch dump happens, the torque from 0ms^-1 to the corresponding velocity at idle engine speed is constant. For our Skyline GTR, the idle engine speed is 950rpm.

Calculation

The basic equation for acceleration is:

which is nothing but Newton's 2nd Law, i.e. mass times acceleration equals to net force. In our case, the net force is the force transmitted from the engine to the tires, subtracting the rolling resistance and air resistance. These are the terms on the right hand side respectively. Then it is rewritten into differential form for integration.

Now the torque curve consists of three parts, it looks like this:

where Γ is in the unit of Nm and Ω is in rpm.

To connect the engine rpm to car speed in m/s, we have the following:

From the results of the previous page, the optimal shift points are:

Gear	1st	2nd
Gear Ratio (g_k)	3.827	2.36
Shift Point (rpm)	8387	7911
rpm at next gear	5172	5648

we have the final torque curve in terms of car speed:

Now we have everything ready for the integration. Plug in all numbers (c_rr=0.015 in this example), we get a set of differential equations:

Integrated from 0 to 100 km/h (=27.8m/s) to get the time:

Conclusion

Under the simplified assumptions, we got 5.41s, this is not very far off from the road test result of 4.9s. If we take into account of rolling resistance, we will have 5.15s which is very close already. Here are some factors that affect our calculation:

We neglect power loss in the torque generated by the engine due to frictions in the mechanical parts. For a efficient sports car, the power loss is about 20%. So for a 280hp rear wheel drive car, the rear wheel horsepower is more like 224hp. This negligence makes our time faster than it should.
We ignore the mass of the driver. Therefore our time is faster than it should.
Shifting gears take time as well. During shifting, no power is transmitted to the wheels. Because of that, our time is faster than it should.
The torque curve used in our calculation is a simplified one. The real curve is more "convex" for the part before the peak. Therefore, the area under the curve, which is related to power, is larger in real situation. However, this is only true in low rpm, after the peak, our torque curve actually gives a higher output than the real one. Since the engine remains at high rpm except from the very beginning, our results should be comparable.
Car magazine drivers usually try to get the most out of their test car. Therefore they usually start the car by dumping the clutch at a high engine speed. We did not model this situation, from our empirical experience, it seems that this can greatly reduce the acceleration time, in particular in the first gear, by 0.5~1s.
Another explanation is that the actual peak power of the GTR exceeds 280hp, some webpages claim it actually has 326hp. This should make our number higher than reality.

As you can see there are many practical factors that contribute to the deviation from the realistic result. Let me leave the task to calculate the perfect number to our intelligent readers! =)

First Draft: November 8th, 2003
1.0 Published: December 12, 2003
Stephen Ng, Yee Man Chan