How to find the minimum time and distance of a sliding friction?

Question

1. The coefficient of sliding friction between a rubber tire and a wet concrete road is 0.5

a. find the minimum time in which a car whose initial velocity is 30 miles/hour can come to a stop on such a road.

b. what distance will that car cover in this time?

Answer 1

Use the friction equation (F_friction = F_normal(μ)) to figure out the (horizontal) force on the car; then use F=ma to figure out the car's acceleration; then use kinematics equations to relate acceleration, speed, distance and time.

F_friction = (F_normal)(μ)

= (weight)(μ)

= (mg)(μ)

If you want to be technical, the friction acts in the opposite direction from the motion, so it probably should have a minus sign to indicate its "negative" direction:

F_friction = -mgμ

The friction is the net force on the car, so use F=ma:

F_friction = ma

Combine the two above equations:

-mgμ = ma

Simplify:

a = -gμ

Part "a":

Use the formula that relates change in speed, acceleration and time:

t = Δv / a

= (0 mph - 30 mph) / (-gμ)

Part "b":

Use the formula that relates speed, acceleration and distance:

d = (v_final² - v_initial²) / (2a)

= ((0mph)² - (30mph)²) / (-2gμ)

CAUTION: Don't forget to convert everything to common units, or your final numbers will come out wrong.

Answer 2

I can only give equations.

F-fk=ma

fk=F-ma

N=W

coefficient of sliding friction=fk/N=(F-ma)/W

You can solve for the distance using the formula S=Vot+1/2at^2 or S=(Vf^2-Vo^2)/2a

time=S/ave. velocity