Question

Question 1 given A car skids 185 ft down a 3% grade to stop. Friction factors are as follows. Vo (mph) 50 40 60 30 0.45 0.51 0.35 0.59 What is the speed at the beginning of the skid? Question 1 given A car skids 185 ft down a 3% grade to stop. Friction factors are as follows. Vo (mph) 50 40 60 30 0.45 0.51 0.35 0.59 What is the speed at the beginning of the skid?

Answer 1

Braking distance (d) = 185 ft

Acceleration due to gravity (g) = 32.2 ft/s²

Descending gradient (n) = 0.03

Using the given data of velocity (v) and friction factor (f), graph is plotted to find the trendline (friction factor as a function of velocity) [Note: velocity is converted from mph to ft/s and used in graph],

0.7 0.6 0.5 0.4 y -3E -06x 3 +0.0006x2 -0.0423x +1.55 0.3 0.2 0.1 0 40 60 100 Velocity (ft/s) 80 20 O Friction factor

f = -3x10^-6v³ + 0.0006v² - 0.0423v + 1.55 (when v is in ft/s)

For descending gradient,

Braking distance (d) = v² / 2g(f-n)

=> 185 = v² / (2 x 32.2 x (-3x10^-6v³ + 0.0006v² - 0.0423v + 1.55 - 0.03))

=> 11914 x (-3x10^-6v³ + 0.0006v² - 0.0423v + 1.52) = v²

=> -0.0357 v³ + 7.148 v² - 503.962 v + 18109.28 = v²

=> -0.0357 v³ + 6.148 v² - 503.962 v + 18109.28 = 0

=> 0.0357 v³ - 6.148 v² + 503.962 v - 18109.28 = 0

=> v = 74.09 ft/s = 50.515 mph

Therefore, speed at beginning of skid = 50.515 mph