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Chapter 4: Problem 5

A police investigation of tire marks showed that a car traveling along a straight and level street had skidded to a stop for a total distance of 200 ft after the brakes were applied. The coefficient of friction between tires and pavement is estimated to be \(\mu=0.7 .\) What was the probable minimum speed (mph) of the car when the brakes were applied? How long did the car skid?

Short Answer

Expert verified
The initial speed when the brakes were applied is approximately 80 mph and the car skidded for about 5 seconds.

Step by step solution

01

Convert units

The given skid distance is in feet (ft) and the speed is required in miles per hour (mph). Assume the mass of the car to be \(m\) kg. Convert the skid distance of 200 ft to a metric measure (m) by multiplying by 0.3048. The converted skid distance would be \(200 \times 0.3048 = 60.96\) m.
02

Calculate initial speed

The initial kinetic energy of the car is given by \(\frac{1}{2}mv^2\), where \(v\) is the initial speed of the car. This is equal to work done by friction in stopping the car, which is the force of friction (mass x gravity x friction coefficient, i.e., \(mg\mu\)) multiplied by the distance (d = 60.96 m). So we get \(\frac{1}{2}mv^2 = mg\mu d\). 'm' cancels out from both sides, giving \(v^2 = 2g\mu d\). Using \(g \approx 9.8\) m/s\(^2\), \(\mu = 0.7\), and \(d = 60.96\) m, we solve this equation to find the initial speed in m/s.
03

Convert units and round off the result

The calculated initial speed needs to be converted back to mph by multiplying by 2.237 (since 1 m/s = 2.237 mph). After converting, round off the result to the nearest integer to get the initial speed in mph.
04

Calculate skid time

The skid time is calculated from the formula \(t = \frac{v}{g\mu}\). Substitute the obtained \(v\) in m/s and calculate the skid time in seconds.

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