Physics of Stop Ramps

While driving on the highway, I noticed that there were periodic ramps of sand off to the side of the road where trucks could roll into to stop if their breaks failed while going down the hill. The ramps seemed rather short, though, which made me start to wonder how the builders know how long to make the ramps so that they are effective at stopping the trucks traveling at highway speed down a hill. I decided to try to figure it out myself…
If an eighteen wheeler has a mass of 36287 kg and is traveling at a speed of 70mph , how long would a sand ramp have to be to stop the truck given its breaks don’t work?  The coefficient of friction of sand is 0.60.

F= ma

F_fr = µF_N = µmg

F_N = mg

µmg = ma

(0.60)(36287kg)(9.8) = (36287kg)a

a = 5.88 m/s²

a = Δv

      Δt

70mph = 31.3 m/s

5.88 m/s² = 31.3 – 0

                              Δt

Δt = 5.32 s

x = v₀t + ½ at²

x = (31.3m/s)(5.32s) + ½ (5.88m/s²)(5.32s)²

x = 250 m

The ramp must be at least 250 m long to stop a truck traveling at 70mph.