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Grade resistance is simply the gravitational force (the component parallel to the roadway) acting on the vehicle. As suggested in Fig. 2.1, the expression for grade resistance (Rg) is

sing gR W θ= (2.8)

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As in the development of the rolling resistance formula (Eq. 2.6), highway grades are usually very small, so sin θg ≅ tan θg. Rewriting Eq. 2.8, we get

tang gR W WGθ≅ = (2.9)


G = grade, defined as the vertical rise per some specified horizontal distance (opposite side of the force triangle, Fig. 2.1, divided by the adjacent side) in ft/ft.

Grades are generally specified as percentages for ease of understanding. Thus a roadway that rises 5 ft vertically per 100 ft horizontally (G = 0.05 and θg = 2.86°) is said to have a 5% grade.


A 2000-lb car has CD = 0.40, Af = 20 ft2, and an available tractive effort of 255 lb. If the car is traveling at an elevation of 5000 ft (ρ = 0.002045 slugs/ft3) on a paved surface at a speed of 70 mi/h, what is the maximum grade that this car could ascend and still maintain the 70-mi/h speed?


To maintain the speed, the available tractive effort will be exactly equal to the summation of resistances. Thus no tractive effort will remain for vehicle acceleration (ma = 0). Therefore, Eq. 2.2 can be written as

a rl gF R R R= + +

For grade resistance (using Eq. 2.9),

2000gR WG G= =

for aerodynamic resistance (using Eq. 2.3),

( )( )( )



2 0.002045

0.4 20 70 5280 / 3600 2

86.22 lb

a D fR C A V ρ=

= ×


and for rolling resistance (using Eq. 2.6),

70 5280 3600 0.01 1 2000


33.97 lb

rl rlR f W= × = + × 

  =


255 86.22 33.97 2000

0.0674 or a 6.74% grade



= = + + =

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