A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are 0.355 and 0.650, respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0 m/s without causing the box to slide? Draw a free-body diagram of the toolbox.

Solution 32E Step 1 In this case the tool box is not sliding, so we will use the static friction. Suppose m is the mass of the tool box. So the maximum frictional force on the toolbox is F = fmg Now the maximum force that can be applied due to acceleration is F = Fa. f Fa F f mg Hence the maximum possible acceleration is a = m = m = m = g We know that = 0.650 2 And g = 9.8 m/s Hence the maximum possible acceleration is a = (0.650)(9.8 m/s ) = 6.37 m/s 2 Now the truck is starting from rest and the final velocity of the truck is v = 30.0 m/s. Now from the dynamics we know that, if an object starts from rest and accelerate for time t, we must have v = at v 30.0 m/s t = a = 6.37 m/s 4.71 s Hence the shortest time that the truck could accelerate uniformly to 30.0 m/s is 4.71 s.