(a) If the coefficient of kinetic friction between tires and...

(a) If the coefficient of kinetic friction between tires and dry pavement is $0.80,$ what is the shortest distance in which you can stop an automobile by locking the brakes when traveling at 28.7 $\mathrm{m} / \mathrm{s}$ (about 65 $\mathrm{mi} / \mathrm{h} ) ?$ (b) On wet pavement the coefficient of kinetic friction may be only $0.25 .$ How fast should you drive on wet pavement in order to be able to stop in the same distance as in part (a)? (Note: Locking the brakes is not the safest way to stop.)

(a) If the coefficient of kinetic friction between tires and dry pavement is $0.80,$ what is the shortest distance in which you can stop an automobile by locking the brakes when traveling at 28.7 $\mathrm{m} / \mathrm{s}$ (about 65 $\mathrm{mi} / \mathrm{h} ) ?$ (b) On wet pavement the coefficient of kinetic friction may be only $0.25 .$ How fast should you drive on wet pavement in order to be able to stop in
the same distance as in part (a)? (Note: Locking the brakes is not the safest way to stop.)

Key Concepts

Kinetic Friction

Kinetic friction is the resistive force that acts between two surfaces in relative motion. It is proportional to the normal force and is characterized by a coefficient (?) which quantifies its magnitude relative to the normal force. In contexts such as braking, kinetic friction between tires and the road is what provides the force necessary to decelerate and eventually stop a vehicle.

Stopping Distance

Stopping distance is the distance required for a moving vehicle to come to a complete stop once the brakes are applied. It is determined by the initial speed of the vehicle and the deceleration provided by the frictional force. The stopping distance can be calculated with the formula d = v²/(2?g), showing that it depends on the square of the vehicle's speed and inversely on the coefficient of friction and gravitational acceleration.

Deceleration due to Friction

When the brakes are applied, the frictional force (given by ?mg, where m is the mass of the vehicle and g is the acceleration due to gravity) causes a constant deceleration. According to Newton’s second law, this deceleration (a) is ?g. This acceleration directly determines how quickly a vehicle can slow down, linking the coefficient of friction to the effectiveness of braking.

Impact of Surface Conditions

Surface conditions such as wet or dry pavement affect the coefficient of kinetic friction. A lower coefficient (like that on wet pavement) results in reduced frictional force and therefore a lower deceleration, which in turn requires adjustments in initial speed to ensure that the vehicle can still stop within a predetermined distance. This illustrates the practical importance of adapting driving speeds to changing road conditions.

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