(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 $29.1 \mathrm{~m} / \mathrm{s}$ (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 surfaces in relative motion. In problems involving stopping distances, the coefficient of kinetic friction quantifies how much frictional force is available to decelerate a moving vehicle when the brakes are locked. Understanding this concept is crucial because it determines the maximum possible frictional force, which in turn affects the vehicle's deceleration and stopping distance.

Deceleration Due to Friction

Deceleration in the context of stopping a moving object is the negative acceleration produced by friction. When brakes are applied, the frictional force (calculated as the product of the coefficient of kinetic friction and the normal force) causes the vehicle to slow down. This deceleration is a key factor in estimating the stopping distance because a higher frictional coefficient leads to a larger deceleration, allowing the vehicle to stop in a shorter distance.

Kinematic Equations of Motion

The kinematic equations for constant acceleration are used to relate speed, acceleration, and stopping distance. One commonly used equation is v^2 = v_0^2 + 2a?x, where v is the final velocity, v_0 is the initial velocity, a is the acceleration (in this case, negative due to deceleration), and ?x is the distance traveled. For stopping problems where the final velocity is zero, these equations allow us to solve for the stopping distance as a function of the initial speed and the deceleration provided by friction.

Quadratic Relationship Between Speed and Stopping Distance

The stopping distance is directly proportional to the square of the speed when deceleration is constant. This quadratic relationship means that even small increases in speed will result in larger increases in stopping distance. This concept is important in understanding why vehicles must travel at reduced speeds under conditions of lower friction, such as on wet surfaces, to maintain the same stopping distance as on dry pavement.

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