Runaway truck ramps are used on mountain roads to stop trucks whose brakes fail.

Question

Runaway truck ramps are used on mountain roads to stop trucks whose brakes fail. The ramps work by using sand to increase the coefficient of kinetic friction between the truck and the road. Suppose that the ramp is flat and is d = 150 m long with a coefficient of kinetic friction mu_k = 4.0. Find the height above the ramp that the brakes of a truck can fail if it is going v_0 = 25 m/s at the time that its breaks fail. You may ignore all sources of friction until the truck reaches the runaway ramp.

Explanation / Answer

work done by the Net force = change in kinetic energy

(m*g*sin(theta) + fk)*d = -0.5*m*Vo^2

fk is the kinetic frictional force = mu_k*m*g*cos(theta) = 0.4*m*9.8*cos(theta)

then

[m*g*sin(theta) +(0.4*m*9.8*cos(theta))]*d = 0.5*m*25^2 -

m cancels on both sides

[(9.8*sin(theta))+(0.4*9.8*cos(theta))]*150 = 0.5*25^2

let sin(theta) = x

then cos(theta) = sqrt(1-x^2)

then

[(9.8*x)+(0.4*9.8*sqrt(1-x^2))]*150 = 0.5*25^2

x = -0.18

sin(theta) = -0.18

theta = sin^(-1)(0.18)

theta = 10.36 degrees

so required height is

h = 150*sin(10.36) = 27 meter

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