A car’s braking distance d (the distance it travels if...

A car’s braking distance d (the distance it travels if rolling to a stop after the brakes are applied) depends on its initial speed $v_{0},$ the maximum friction force $\vec{f}_{\mathrm{R} \text { on } \mathrm{C}}$ exerted by the road on the car, and the car's mass $m$ according to the equation $$\frac{2 f_{s \max }}{m} d=v_{0}^{2}$$ Suppose the braking distance for a particular car and road surface is 26 $\mathrm{m}$ when the initial speed is 18 $\mathrm{m} / \mathrm{s} .$ What is the braking distance when traveling at 27 $\mathrm{m} / \mathrm{s} ?$ (a) 59 m (b) 39 m (c) 26 m (d) 17 m (e) 12 m

A car’s braking distance d (the distance it travels if rolling to a stop after the brakes are applied) depends on its initial speed $v_{0},$ the maximum friction force $\vec{f}_{\mathrm{R} \text { on } \mathrm{C}}$ exerted by the road on the car, and the car's mass $m$ according to the equation
$$\frac{2 f_{s \max }}{m} d=v_{0}^{2}$$
Suppose the braking distance for a particular car and road surface is 26 $\mathrm{m}$ when the initial speed is 18 $\mathrm{m} / \mathrm{s} .$ What is the braking distance when traveling at 27 $\mathrm{m} / \mathrm{s} ?$
(a) 59 m (b) 39 m (c) 26 m
(d) 17 m (e) 12 m

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Key Concepts

Proportional Relationship

Braking distance is directly proportional to the square of the initial speed when other variables, such as friction and mass, remain constant. This means that any change in speed will have an amplified effect on the stopping distance, making understanding quadratic scaling essential in vehicle dynamics and safety analysis.

Friction and Deceleration

Friction provides the decelerating force that stops the vehicle. The maximum friction force, coupled with the mass of the car, determines the rate of deceleration, and therefore directly influences the braking distance. This concept connects the physical properties of the tire-road interface to the mechanics of motion.

Work-Energy Principle

The relationship between braking distance and the square of the initial speed can be derived using the work-energy principle. The work done by the friction force over the braking distance is equal to the initial kinetic energy of the car, which emphasizes how energy considerations underlie the motion and the stopping process.

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