Question

A truck is traveling down a road with a 4-percent grade at a speed of 75 mi/h when its brakes are applied to slow it down to 22.5 mi/h. An antiskid braking system limits the braking force to a value at which the wheels of the truck are just about to slide. Knowing that the coefficient of static friction between the road and the wheels is 0.60, determine the shortest time needed for the truck to slow down.

Answer 1

3.964 s

Step-by-step explanation:

Metric unit conversion:

1 miles = 1.6 km = 1600 m.

1 hour = 60 minutes = 3600 seconds

75 mph = 75 * 1600 / 3600 = 33.3 m/s

22.5 mph = 22.5 * 1600/3600 = 10 m/s

Let g = 9.81 m/s2

Friction is the product of coefficient and normal force, which equals to the gravity

`F_f = \mu N = \mu mg`

The deceleration caused by friction is friction divided by mass according to Newton 2nd law.

`a = F_f / m = \mu mg / m = \mu g = 0.6 *9.81 = 5.886 m/s^2`

So the time required to decelerate from 33.3 m/s to 10 m/s so the wheels don't slide, with the rate of 5.886 m/s2 is

`t = (\Delta v)/(a) = (33.3 - 10)/(5.886) = 3.964 s`