Library / Automotive and vehicle dynamics / VEH-04
Automotive and vehicle dynamics
Vehicle acceleration from road load
Straight-line acceleration from the force balance: drive force minus aerodynamic drag, rolling resistance and the weight component along the road, divided by the equivalent mass.
- Ref
- VEH-04
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- a [m/s²]
Fdrive is the force from the driven wheels, ½ρCDAU² is the aerodynamic drag, Crr m g cos θ the rolling resistance and m g sin θ the weight component along the road (θ positive uphill, negative downhill). The net force divided by the equivalent mass ε m gives the acceleration. Wind is taken as zero, so the air speed equals the vehicle speed U. If the result is negative, the vehicle slows down at that speed.
Straight-line acceleration from the force balance: drive force minus aerodynamic drag, rolling resistance and the weight component along the road, divided by the equivalent mass. θ is positive uphill.
Calculator
- Aerodynamic drag, ½ρCDAU²158.4 N
- Rolling resistance, Crr m g cos θ220.7 N
- Weight component along the road, m g sin θ0 N
- Net force on the vehicle5,621 N
- Acceleration in g0.3537 g
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 the defaults the resistances (379.1 N) are about 6.3 % of the drive force, so most of it goes into acceleration: 3.47 m/s², about 0.35 g.
- A 10 % uphill grade (5.7°) cuts the acceleration to 2.567 m/s² with the same drive force, because the weight component alone takes about 1,464 N.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Vehicle mass | 500 to 5,000 kg | 0.9455 | 10.68 | 9.736 | +10% | −9.45 % |
| Fdrive Wheel drive force | 0 to 15 kN | −0.234 | 9.025 | 9.259 | +10% | +10.7 % |
| θ Road gradient angle (positive uphill) | −15 to 15 ° | 1.123 | 5.825 | 4.702 | +0.6 ° | −2.74 % |
| ε Equivalent mass factor, m_{eq}/m | 1 to 1.5 | 2.498 | 3.747 | 1.249 | +10% | −9.09 % |
| U Vehicle speed | 0 to 60 m/s | 2.687 | 3.567 | 0.88 | +10% | −0.592 % |
| Crr Rolling resistance coefficient | 0.005 to 0.1 | 2.698 | 3.561 | 0.8629 | +10% | −0.393 % |
| A Frontal area | 1 to 8 m² | 3.212 | 3.523 | 0.3111 | +10% | −0.282 % |
| CD Drag coefficient | 0.2 to 0.8 | 3.307 | 3.502 | 0.1956 | +10% | −0.282 % |
| ρ Air density | 0.8 to 1.3 kg/m³ | 3.462 | 3.502 | 0.04074 | +10% | −0.282 % |
Sweep of Vehicle speed
| Vehicle speed (m/s) | Acceleration (m/s²) |
|---|---|
| 0 | 3.567 |
| 7.5 | 3.554 |
| 15 | 3.512 |
| 22.5 | 3.444 |
| 30 | 3.347 |
| 37.5 | 3.224 |
| 45 | 3.072 |
| 52.5 | 2.894 |
| 60 | 2.687 |
Sweep one variable
Variables and how each one contributes
Acceleration
What the equation gives you. Everything else feeds it.
Wheel drive force
The force pushed forward by the driven wheels, in kN. It comes from the drive-force equation, limited by grip. Adds directly to the net force, so the acceleration rises faster than the drive force. Raising it by 10 % (to 6.6) moves the result by +10.7 %, from 3.47 to 3.84 m/s².
Vehicle speed
The vehicle speed, in m/s. Aerodynamic drag grows with its square. Raises the drag with U², which lowers the acceleration slightly here but becomes decisive at high speed. Raising it by 10 % (to 22) moves the result by −0.592 %, from 3.47 to 3.449 m/s².
Vehicle mass
The vehicle mass, in kg. Raises the rolling resistance and the inertia, so the vehicle accelerates less. Raising it by 10 % (to 1,650) moves the result by −9.45 %, from 3.47 to 3.142 m/s².
Equivalent mass factor, m_{eq}/m
The equivalent mass factor meq/m, which allows for the energy needed to spin up the wheels and engine. It is about 1.04 to 1.1 in top gear and more in low gears. Divides the acceleration directly, so a 10 % larger factor gives about 10 % less acceleration. Raising it by 10 % (to 1.188) moves the result by −9.09 %, from 3.47 to 3.154 m/s².
Air density
The air density, in kg/m³: 1.2 at sea level and 20 °C, less at altitude or in heat. Scales the drag. At 20 m/s the drag is a small part of the total, so the effect is small. Raising it by 10 % (to 1.32) moves the result by −0.282 %, from 3.47 to 3.46 m/s².
Drag coefficient
The drag coefficient, a dimensionless measure of how streamlined the vehicle is: 0.25 to 0.35 for a car, 0.5 to 0.8 for a box-shaped vehicle. Scales the drag, a small effect at this speed. Raising it by 10 % (to 0.33) moves the result by −0.282 %, from 3.47 to 3.46 m/s².
Frontal area
The frontal area of the vehicle, in m². Scales the drag, a small effect at this speed. Raising it by 10 % (to 2.42) moves the result by −0.282 %, from 3.47 to 3.46 m/s².
Rolling resistance coefficient
The rolling resistance coefficient: about 0.01 to 0.015 on good tarmac, 0.04 or more on gravel and 0.1 or more on soft sand. Scales the rolling resistance, which is small at the default value on tarmac. Raising it by 10 % (to 0.0165) moves the result by −0.393 %, from 3.47 to 3.456 m/s².
Road gradient angle (positive uphill)
The road gradient angle in degrees, positive uphill and negative downhill. A 10 % grade is about 5.7°. Each degree of uphill gradient adds about 0.017 m g of resistive force, so the gradient is a powerful lever. Raising it by 2° (to 2°) moves the result by −9.13 %, from 3.47 to 3.153 m/s².
Worked example with the values above
- Resistances: drag = 158.4 N, rolling = 220.7 N, grade = 0 N, total 379.1 N.
- Net force: 6,000 − 379.1 = 5,621 N.
- a = net force / (ε m) = 5,621 / (1.08 × 1,500) = 3.47 m/s².
Assumptions and limits
- Straight-line motion on a uniform grade, in still air, so the air speed equals the vehicle speed.
- Constant drag and rolling coefficients, a constant equivalent mass factor, and the drive force known at this speed.
Common mistakes
- Using the plain mass in place of the equivalent mass. Rotating inertia typically adds 5 to 10 % in top gear and much more in first.
- Entering a downhill slope as a positive angle. θ is positive uphill, so a downhill slope is negative.
Statistics and uncertainty
The acceleration is a small difference of forces divided by mass. At low speed the drive force dominates, so a 20 % error in the drag coefficient changes a by less than 1 %. Near top speed the drive force and the resistances almost cancel, and the small remainder becomes very sensitive: a 5 % error in the drag or in the drive force can change the acceleration by tens of per cent, and the top speed itself is uncertain by a few per cent.
Where it is used
- Estimating 0–100 km/h time by stepping the equation over speed.
- Finding the top speed, where the acceleration falls to zero, and the steepest hill a given drive force can climb.
Maths you need
- Newton’s second law: F = m a
- Resolving forces on a slope: weight components m g sin θ and m g cos θ