Problem 2. The minimum safe stopping distance (S) for the vehicles on level road

Question

Problem 2. The minimum safe stopping distance (S) for the vehicles on level road depends on the speed (v) at the beginning of the deceleration and the coefficient of friction (f) between the tires and the pavement. The following relationship is suggested to estimate the stopping distance for wet pavement: S = avbfc Using the following 17 sets of data, estimated a, b and c v (mph) 10 15 20 25 30 35 40 45 48 18 38 S (ft) 43 80 74 125 170 190 185 35 185 95 65 160 110 215 100 0.40 0.36 0.29 0.27 0.40 0.33 0.31 0.36 0.40 0.33 0.27 0.36 0.40 0.36 0.40 0.40 0.31 28 42 36 50 30

Explanation / Answer

Here

S = avb fc

ln S = ln a + b ln v + c ln f

so we will take logarithm of values for v, f and S and then do a multiple regression with least square regression method.

The regression output is

ln S = -3.1294 + 1.96468 ln v - 0.90953 ln f

so here coefficients are

ln a = -3.1294

a = 0.0437

b = 1.96468

c = - 0.90953

S = 0.0437 v1.96468  f-0.90953

SUMMARY OUTPUT Regression Statistics Multiple R 0.972855 R Square 0.946447 Adjusted R Square 0.938796 Standard Error 0.214847 Observations 17 ANOVA df SS MS F Significance F Regression 2 11.42083 5.710413 123.7114 1.26E-09 Residual 14 0.646228 0.046159 Total 16 12.06705 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -3.12941 0.59825 -5.23095 0.000127 -4.41253 -1.84629 ln v 1.96468 0.125748 15.62394 2.96E-10 1.694977 2.234383 ln f -0.90953 0.382918 -2.37527 0.032364 -1.73081 -0.08826

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