Library / Automotive and vehicle dynamics / VEH-35
Automotive and vehicle dynamics
Power for steady level-road speed
Power needed at the wheels to hold a constant speed on level ground in still air: the aerodynamic drag force plus the rolling resistance force, multiplied by the speed.
- Ref
- VEH-35
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- Pw [kW]
The steady-speed condition on level ground in still air balances the drive force against the aerodynamic drag ½ρCDAU² and the rolling resistance Crr m g. The power at the wheels is that total force times the speed U. The result excludes the drivetrain losses (divide by ηd to get the engine power) and the power for accessories. Compare it with the engine power available at that speed, and check the engine speed limit in the chosen gear.
Power needed at the wheels to hold a constant speed on level ground in still air: the aerodynamic drag force plus the rolling resistance force, multiplied by the speed.
Calculator
- Aerodynamic drag power, ½ρCDAU³52.5 kW
- Rolling resistance power, Crr m g U11.04 kW
- Share of the power spent on drag82.63 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 50 m/s (180 km/h) the car needs 63.54 kW at the wheels, of which 82.6 % goes on aerodynamic drag. At half the speed (25 m/s) it needs only 12.08 kW.
- The drag power rises with the cube of the speed, so a 26 % increase in speed needs about twice the power. This is the reason that top speed is limited by power, and that cruising fuel consumption rises sharply above about 100 km/h.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kW) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U Speed | 5 to 90 m/s | 1.156 | 326 | 324.9 | +10% | +29.1 % |
| A Frontal area | 1 to 8 m² | 37.29 | 221 | 183.8 | +10% | +8.26 % |
| CD Drag coefficient | 0.2 to 0.8 | 41.04 | 131 | 90 | +10% | +8.26 % |
| Crr Rolling resistance coefficient | 0.005 to 0.1 | 56.18 | 126.1 | 69.9 | +10% | +1.74 % |
| m Vehicle mass | 500 to 5,000 kg | 56.18 | 89.29 | 33.11 | +10% | +1.74 % |
| ρ Air density | 0.8 to 1.3 kg/m³ | 46.04 | 67.91 | 21.88 | +10% | +8.26 % |
Sweep of Speed
| Speed (m/s) | Wheel power (kW) |
|---|---|
| 5 | 1.156 |
| 15.63 | 5.051 |
| 26.25 | 13.39 |
| 36.88 | 29.2 |
| 47.5 | 55.5 |
| 58.13 | 95.31 |
| 68.75 | 151.7 |
| 79.38 | 227.6 |
| 90 | 326 |
Sweep one variable
Variables and how each one contributes
Wheel power
What the equation gives you. Everything else feeds it.
Air density
The air density, in kg/m³. Directly proportional to the drag power, so high altitude or hot weather reduces the power needed. Raising it by 10 % (to 1.32) moves the result by +8.26 %, from 63.54 to 68.79 kW.
Drag coefficient
The drag coefficient, a dimensionless measure of the shape: 0.25 to 0.35 for a car. Directly proportional to the drag power. At this speed it dominates the total. Raising it by 10 % (to 0.385) moves the result by +8.26 %, from 63.54 to 68.79 kW.
Frontal area
The frontal area of the vehicle, in m². Directly proportional to the drag power. Raising it by 10 % (to 2.2) moves the result by +8.26 %, from 63.54 to 68.79 kW.
Rolling resistance coefficient
The rolling resistance coefficient: about 0.01 to 0.015 for road tyres on tarmac. Directly proportional to the rolling power, which is a small part of the total at this speed. Raising it by 10 % (to 0.0165) moves the result by +1.74 %, from 63.54 to 64.64 kW.
Vehicle mass
The vehicle mass, in kg. Affects only the rolling power, so it has a small effect at high speed. Raising it by 10 % (to 1,650) moves the result by +1.74 %, from 63.54 to 64.64 kW.
Speed
The steady speed, in m/s. The drag power rises with the cube of the speed, so speed is by far the strongest lever at the top end. Raising it by 10 % (to 55) moves the result by +29.1 %, from 63.54 to 82.02 kW.
Worked example with the values above
- Drag force: ½ρCDAU² = 0.5 × 1.2 × 0.35 × 2 × 50² = 1,050 N. Rolling force: Crr m g = 0.015 × 1,500 × 9.81 = 220.7 N.
- Total force: 1,271 N. Multiply by the speed: 1,271 × 50 = 63,540 W.
- Pw = 63.54 kW at the wheels.
Assumptions and limits
- Steady speed on a level road in still air, so the air speed equals the vehicle speed.
- Constant drag and rolling coefficients (rolling resistance in reality rises slowly with speed).
Common mistakes
- Using this wheel power as engine power. Divide by the drivetrain efficiency.
- Forgetting the cube of the speed and scaling the power linearly with speed.
Statistics and uncertainty
The drag term depends on the product ρ CD A, whose combined uncertainty is typically 5 to 8 %, and on the speed cubed, so a 2 % speedometer error gives about 6 % in the drag power. Rolling resistance is uncertain to 10 to 20 % but is a minor part of the total at speed. Wind and road slope, which this formula leaves out, can easily make larger differences.
Where it is used
- Estimating the top speed of a vehicle for a given engine power.
- Sizing the motor for a target cruising speed and estimating cruising range.
Maths you need
- Power: P = F U
- Drag force: ½ρCDAU²