Friction and stopping distance of a truck

Question

A 50-kg box is placed in the bed of a truck. The coefficient of static friction between the box and the truck bed is 0.54. If the truck is traveling at 100 km/h then calculate the minimum distance the truck can stop in without the box sliding into the cab.

Answer 1

m = mass of box

f = static friction

 

Find the static friction on the box.  The horizontal force applied by the truck while stopping must not exceed the static friction.  The static friction must balance the force the truck applies while stopping. 

 

f = ma_max 

 

From kinematics, the magnitude of the acceleration of the truck is

 

a = v² / (2d) ⇒

 

a_max = v²/(2d_min)

 

d_min = v² / (2a_max)

 

f = ma_max = μ_smg ⇒

 

m[v²/(2d_min)] = μ_smg ⇒

 

d_min = v² / (2μ_sg)

 

v = 100 km/h = (100,000 m)/(3600 s) ≅ 27.78 m/s 

 

d_min = v² / (2μ_sg) = (27.78)² / [2(0.54)(9.8)] m ≅ 72.91 m