[To understand the material of these documents, I assume the reader have a firm understanding of Newtonian Kinematics.]
For people driving cars on the public road, the maximum speed of a car probably doesn't mean much for them unless they enjoy the feeling of getting caught speeding. People live in Montana probably don't have this concern because there are freeways that don't have speed limit. So now suppose you are shopping for a car for a trip to one of those freeways or a race track somewhere to get the thrill of your life, you definitely want to find the best deal for the speed. Then you need to know how to calculate the maximum speed of a car.
| Γ(ω)Ggk | 1 | ||
| ------------ | = | -- | cdAρv2 |
| r | 2 |
Now multiply both sides by v where v also equals 2πrω/Ggk. Then we have:
| Γ(ω)Ggk | 2πrω | 1 | |||
| ------------- | × | ------- | = | -- | cdAρv2 |
| r | Ggk | 2 |
Note thath Power P = 2πΓ(ω)ω, so
| ( | 2P | ) | ^(1/3) | ||
| v | = | ( | ------- | ) | |
| ( | cdAρ | ) |
Note that when P is at its peak, v is also at its peak. So when we plug in peak power for P, we get the top speed. For our Skyline, this is 262.33km/h.
In reality, the maximum speed is limited by the gearing. Here is why: For each gear, there is a correspondence between engine speed and the speed of the car. Given the top speed occurs only at the peak power that is only attained at a particular engine speed. If at that peak power engine speed, the corresponding car speed is not the maximum speed in any gear, that means the theoretical maximum speed can't be achieved.
This happens because we enter the declining part of our torque curve and at a certain engine speed the force generated by the engine is equal to the air resistance. But since our torque curve is declining, there is no way we can overcome the air resistance to attain higher speed. This holds true in every gear. But as we know when we shift up the gears, we get substantially lower torque from our engine and hence it is more likely that the force generated by the engine can be canceled out by the air resistance. To see whether we can attain the ideal top speed for each gear, we need to look at the force generated by engine at red line and compare it with the air resistance.
| Gear | 1st | 2nd | 3rd | 4th | 5th | 6th | reverse |
| Gear Ratio | 3.827 | 2.36 | 1.685 | 1.312 | 1 | 0.793 | 3.28 |
| Peak Engine Force (N) | 16283.38 | 10041.49 | 7169.45 | 5582.39 | 4254.87 | 3374.11 | 13955.97 |
| Air Resistance at Torque Peak (N) | 65.5 | 172.25 | 337.85 | 557.33 | 959.36 | 1525.58 | 89.17 |
| Engine Force at Red Line (N) | 8994.56 | 5546.69 | 3960.24 | 3083.58 | 2350.29 | 1863.78 | 7708.95 |
| Air Resistance at Top Speed (N) | 244.45 | 642.82 | 1260.99 | 2079.91 | 3580.25 | 5693.33 | 332.79 |
As we can see, it is impossible to attain top speed in the 5th and the 6th gear for our Skyline. To see the top speed attainable in these two gears, we need to see at which point the engine force cancels out the air resistance.
| Γ(ω)Ggk | 1 | ||
| ------------ | = | -- | cdAρv2 |
| r | 2 |
For our Skyline, this condition is equivalent to:
| (-2.568ω+580.32)Ggk | 1 | ||
| ------------------------- | = | -- | cdAρv2 |
| r | 2 |
Recall that v=2πrω/Ggk, we can transform the condition into a quadratic equation in terms of v.
| 1 | 2.568G2gk2 | 580.32Ggk | ||||||
| -- | cdAρv2 | + | ------------ | v | − | ------------ | = | 0 |
| 2 | 2πr2 | r |
The quadratic equation for the 5th gear is:
The quadratic equation for the 6th gear is:
To summarize:
| Gear | 5th | 6th |
| Gear Ratio | 1 | 0.793 |
| Drag Limited Top Speed (km/h) | 261.28 | 261.05 |
| Drag Limited Engine Speed (RPM) | 7522.69 | 5960.25 |
| Corresponding Engine Force (N) | 2804.28 | 2799.35 |
| Corresponding Drag Force (N) | 2804.28 | 2799.35 |
As we can see, in reality, our Skyline can never achieve the theoretical top speed 262.33km/h under the current gearing. The top speed 261.28km/h is attained at the 5th gear. This is not too far off the bat. However, if the gearing designer did a bad job, the car may never get close to its theoretical top speed.
The result of this section also invalidates the optimal shift point we found in the last section for the 5th gear. In fact, we can never attain that shift point, we reach the maximum speed right before it.