### Definition

The road slope (gradient) force is the force which is acting onto the vehicle when it’s rolling on a road with a particular slope (gradient). The road slope force appears due to the decomposition of the vehicle’s weight force on two directions:

• longitudinal axis of the vehicle: road slope force
• vertical axis (normal) of the vehicle

The road slope force acts differently, depending on the sign of the slope:

• positive road slope (climbing): the slope force is resistive to the vehicle movement, it tries to slow down the vehicle
• negative road slope (descending): the slope force accelerates the vehicle

Depending on country or industry, the road slope is also called gradient, incline or grade but in the end it refers to the same physical feature of the road.

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### Formula

Assuming we have a road with the slope angle θ [°], if we decompose the weight force W [N] of the vehicle along the longitudinal and vertical axes of the vehicle, we get the expression of the road slope force Frs [N] as:

Frs = W · sin(θ) = m · g · sin(θ)
(1)

where m [kg] is the vehicle mass and g [m/s2] is the gravitational acceleration.

The road slope power Prs [W] can be calculated by multiplying the road slope force Frs [N] with the vehicle speed v [m/s]:

Prs = Frs · v = m · g · sin(θ) · v
(2)

The road slope power is the amount of power at the driving wheels needed for the vehicle to be able to climb a hill with the slope θ [°].

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The slope of a particular road can be expressed in several way:

• as a percentage, for example 4%
• as a ratio between “rise” and “run”, for example 1:20, where 1 is the “rise” and 20 is the “run”.
• as an angle in radians, for example 5 °

In order to calculate the road slope force and power, we need to have the slope angle in radians.

In the image above the road slope is 10%. This percentage comes from the ratio between “rise” and “run” multiplied by 100.

s [%] = (y/x) · 100
(3)

To calculate the angle, we can use the arc tangent function:

(4)

The angle can be converted from radians into degrees as:

θ [°] = θ [rad] · 180/π
(5)

For small angles, the tangent and sine functions are nearly equal, which means that we can replace the sine function with the actual slope value, given by the ratio between “rise” and “run”.

sin(θ) = tan (θ) = y/x = s [%] / 100
(6)

Equation (6) can be used in embedded software applications in order to avoid implementing trigonometric functions. According to [1], for slopes less than 20%, using this approximation gives an error smaller than 2%.

In the table below you can see the different values of the road slope in different formats.

 Road slope [rise:run] Road slope [%] Slope angle [rad] Slope angle [°] sin(Slope angle) [-] Road slope [rise/run] Approximation Error [%] 1:1 100 0.7854 45 0.7071 1 41.42 1:2 50 0.4636 26.57 0.4472 0.5 11.8 1:3 33.33 0.3218 18.43 0.3162 0.3333 5.41 1:4 25 0.245 14.04 0.2425 0.25 3.08 1:5 20 0.1974 11.31 0.1961 0.2 1.98 1:6 16.67 0.1651 9.46 0.1644 0.1667 1.38 1:7 14.29 0.1419 8.13 0.1414 0.1429 1.02 1:8 12.5 0.1244 7.13 0.124 0.125 0.78 1:9 11.11 0.1107 6.34 0.1104 0.1111 0.62 1:10 10 0.0997 5.71 0.0995 0.1 0.5 1:11 9.09 0.0907 5.19 0.0905 0.0909 0.41 1:12 8.33 0.0831 4.76 0.083 0.0833 0.35 1:13 7.69 0.0768 4.4 0.0767 0.0769 0.3 1:14 7.14 0.0713 4.09 0.0712 0.0714 0.25 1:15 6.67 0.0666 3.81 0.0665 0.0667 0.22 1:16 6.25 0.0624 3.58 0.0624 0.0625 0.2 1:17 5.88 0.0588 3.37 0.0587 0.0588 0.17 1:18 5.56 0.0555 3.18 0.0555 0.0556 0.15 1:19 5.26 0.0526 3.01 0.0526 0.0526 0.14 1:20 5 0.05 2.86 0.0499 0.05 0.12 1:21 4.76 0.0476 2.73 0.0476 0.0476 0.11 1:22 4.55 0.0454 2.6 0.0454 0.0455 0.1 1:23 4.35 0.0435 2.49 0.0434 0.0435 0.09 1:24 4.17 0.0416 2.39 0.0416 0.0417 0.09 1:25 4 0.04 2.29 0.04 0.04 0.08 1:26 3.85 0.0384 2.2 0.0384 0.0385 0.07 1:27 3.7 0.037 2.12 0.037 0.037 0.07 1:28 3.57 0.0357 2.05 0.0357 0.0357 0.06 1:29 3.45 0.0345 1.97 0.0345 0.0345 0.06 1:30 3.33 0.0333 1.91 0.0333 0.0333 0.06 1:31 3.23 0.0322 1.85 0.0322 0.0323 0.05 1:32 3.13 0.0312 1.79 0.0312 0.0313 0.05 1:33 3.03 0.0303 1.74 0.0303 0.0303 0.05 1:34 2.94 0.0294 1.68 0.0294 0.0294 0.04 1:35 2.86 0.0286 1.64 0.0286 0.0286 0.04 1:36 2.78 0.0278 1.59 0.0278 0.0278 0.04 1:37 2.7 0.027 1.55 0.027 0.027 0.04 1:38 2.63 0.0263 1.51 0.0263 0.0263 0.03 1:39 2.56 0.0256 1.47 0.0256 0.0256 0.03 1:40 2.5 0.025 1.43 0.025 0.025 0.03 1:41 2.44 0.0244 1.4 0.0244 0.0244 0.03 1:42 2.38 0.0238 1.36 0.0238 0.0238 0.03 1:43 2.33 0.0233 1.33 0.0232 0.0233 0.03 1:44 2.27 0.0227 1.3 0.0227 0.0227 0.03 1:45 2.22 0.0222 1.27 0.0222 0.0222 0.02 1:46 2.17 0.0217 1.25 0.0217 0.0217 0.02 1:47 2.13 0.0213 1.22 0.0213 0.0213 0.02 1:48 2.08 0.0208 1.19 0.0208 0.0208 0.02 1:49 2.04 0.0204 1.17 0.0204 0.0204 0.02 1:50 2 0.02 1.15 0.02 0.02 0.02

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### Example

A vehicle of mass 2000 kg needs to climb a road with the slope of 1:3. Calculate the required wheel force. What is the required wheel power in order to move the vehicle at a speed of 10 kph?

Step 1. Calculate the road slope angle using equations (4).

θ = atan(1/3) = 0.32175 rad

Step 2. Calculate the road slope force using equation (1).

Frs = 2000 · 9.81 · sin(0.32175) = 6204 N

Step 3. Calculate the road slope power using equation (2).

Prs = 6204 · (10/3.6) = 17234 W = 17.234 kW

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### Calculator

 Vehicle mass [kg] = Road slope [%] = Vehicle speed [kph] = Frs [N] = Prs [kW] = Road slope [°] =

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### References

[1] Automotive Handbook, 9th Edition, Bosch, 2014.