[To understand the material of these documents, I assume the reader have a firm understanding of Newtonian Kinematics.]
I bought a '93 Mazda Protege DX in September 1999. This is my first car. Back then, I didn't know a lot about cars. I also didn't want to pay a lot for a car that ultimately may not be the kind of car I want, so I didn't follow the pack to buy a new Camry or a new Accord.
My original plan was to buy a new car two years later. I wished I will have enough knowledge and also money to buy the kind of car I want by that time. It turns out to be quite true.
I had more free time recently, so I spent some time to study many different kind of cars. When I was studying cars, I met many unfamiliar terms in the car magazines and car specifications. To understand these terms and to interpret them correctly, I was inevitably drawn into the physics of cars by my intellectual curiosity.
While there are many aspects of the car that involves physical laws, since cars are mostly used to commute from point A to point B, I am only going to talk about the kinematcs (or the physics of motion) of cars.
Everybody knows that the engine of a car moves the car forward. However, not too many people realize that the engine only generates force and hence acceleration. Newton's First Law dictates that the inertia of a car alone keeps it moving forward at a constant speed when the net force on the car is zero. Newton's Second Law dictates that when the net force on the car is negative, the car decelerates; if it is positive, the car accelerates.
To put Newton's Laws in perspective, let me introduce the formula to calculate the net force acting on a car:
| Γ(ω)Ggk | 1 | |||||||
| F | = | ------------ | – | crrmg | – | -- | cdAρv2, | where |
| r | 2 |
| F | = | net force acting on the car (N) |
| Γ(ω) | = | torque function of the engine given an engine speed ω in s-1 (Nm) |
| G | = | final drive ratio (no unit) |
| gk | = | k-th gear ratio (no unit) |
| r | = | radius of tire (m) |
| crr | = | rolling resistance coefficient (no unit) |
| m | = | mass of the car (kg) |
| g | = | gravity (ms-2) |
| cd | = | drag (air resistance) coefficient (no unit) |
| A | = | frontal area of the car (m2) |
| ρ | = | density of air (kgm-3) |
| v | = | velocity of car (ms-1) |
The first component of the formula is the force exerted by the engine of the car. The second component is the rolling resistance force experienced by the car due to the friction between the tires and the road. The last component is the air resistance force acting on the front of the car when the car sift through the air. There are many other components in this formula. But most of them are insignificant and hence they are dropped. I am also opt for dropping the rolling resistance because I can't find any reliable source for the rolling resistance coefficient and I want to keep the calculations simple. Hence, unless otherwise specified, the rolling resistance force will be dropped from the formula throughout the discussion.
To illustrate the theory, I need to use a car as an example. The car of my choice is 2001 Nissan Skyline GT-R R34 V-spec II N1. Many of you may not even know this car. It is understandable because this car is never imported to North America by Nissan. But if you are a car enthusiast, you will know that it is one of the best GT (Grand Touring) cars in the world. Originally, I would like to pick Honda S2000 but unfortunately, only the Skyline has a peaky enough torque curve to facilitate the discussion of optimal shift point. (Well, the real torque curve may not be that peaky. It is because production cars in Japan can only report 206000W power at most according to the industry consensus. Rumors have that the Skyline is a 300-plus horsepower machine! [1 horsepower = 745W])
Anyway, here is the specification of 2001 Nissan Skyline GT-R R34 V-spec II N1 I got from Nissan's Japanese web site:
air density = 1.29kgm-3
mass = 1550kg
red line = 8500rpm
drag coefficient (cd) = 0.34 (not an official number)
height = 1.36m
width = 1.785m
frontal area = height × width = 2.4276m2 (only an approximation)
wheelbase = 2.665m
tire = 245/40ZR18
tire radius = 0.3266m (this number is calculated from the tire spec)
torque = 392Nm@4400rpm
power = 206000W@6800rpm
1st gear ratio = 3.827
2nd gear ratio = 2.36
3rd gear ratio = 1.685
4th gear ratio = 1.312
5th gear ratio = 1
6th gear ratio = 0.793
reverse gear ratio = 3.28
final drive ratio = 3.545
lateral acceleration = 0.94g on skidpad of 300ft radius (From Road & Track magazine Feb 1999)
weight distribution (front/rear) = 57%/43% (From Road & Track magazine Feb 1999)
minimum turning radius = 5.6m
While most entries in the specification are self-explanatory, there are still some of them I need to clarify.
When I was reading car magazines, the articles usually touted about the horsepower of a specific car. It seems to an average reader that the more horsepower a car has, the more powerful it is and the faster it is. In a sense, they are somehow related. But it is misleading at best.
There are several misleading points here. First the horsepower they quote is the peak horsepower at a certain engine speed. That means in any other engine speed, the horsepower is lower than that value. Obviously, you are not going to stay at that horsepower in most of the time.
Another thing you need to think about is what it means to be fast for you. If you are looking for the maximum speed of a car, there are three numbers you can look at: high horsepower, low drag coefficient or low frontal area. [Gearing is also an important topic but it is more advanced, so I will leave it to the Maximum Speed section.] I will have a closer look at these factors in the Maximum Speed section. But if you are looking for best acceleration at low speed, you should look into car weight, gearing and torque curve. If you are looking for best acceleration in high speed, you also need to take care of the drag coefficient and the frontal area as well.
The thing seems to be quite complicated. But that's why these documents are for. Right? ^_~
So suppose you are more concerned about the acceleration, you definitely need to understand torque. In car engineering, torque is the force geneated by the engine motor multiplied by the radius of the motor arm. After going through the transmission (ie the gears) and reach the tire, torque becomes force that acts on the tires backward which in turn accelerates the car forward according to Newton's Third Law. Recall our force formula, then you will see how torque translates into force and hence acceleration.
Finally, to your surprise, power and torque are related. They are related in the following formula:
| P | = | 2πΓ(ω)ω, where |
| P | = | Power generated by engine (Js-1) |
| Γ(ω) | = | Torque generated by engine at engine speed ω (Nm) |
| ω | = | engine speed (s-1) |
In the Optimal Shift Point section, I will use this relationship to sketch a simple torque curve. From the formula, we also see that, the more horsepower you have, the more torque you have. So in a sense, high horsepower also gives you higher acceleration, although they are not proportional in any sense.
When you look at the wall of your car's tire, you will find out there are some funny numbers written on it. In our Skyline, it says 245/40ZR18.
These numbers and alphabets described information regarding the tire. The first number before '/' is the width of the tire in millimeters. The second number after '/' is the aspect ratio of the tire, ie tire thickness in terms of the percentage of tire width. The last number is the diameter of the wheel this tire fits in inches. So by adding the numbers up, we know the tire radius is (245*0.4/1000+0.5*18*2.54/100) = 0.3266m
The alphabets tell you the maximum speed the tire can handle. E=70km/h, F=80km/h, G=90km/h, J=100km/h, K=110km/h, L=120km/h, M=130km/h, N=140km/h, P=150km/h, Q=160km/h, R=170km/h, S=180km/h, T=190km/h, U=200km/h, H=210km/h, V=240km/h, W=270km/h, Y=300km/h, ZR=greater than 240km/h. For our Skyline, it is of course using the ZR ones.
First Draft: November 18th, 2000
1.0 Published: November 21st, 2000
Yee Man Chan