Answer:
58.7%
Step-by-step explanation:
First, before calculating the braking efficiency of the car; we need to find the coefficient of the rolling friction of the driving by applying the formula;
[tex]f_r_l[/tex] [tex]= 0.01(1+\frac{(\frac{V_1+V_2)}{2} }{147})[/tex]
where;
V₁ = 90 mi/h
V₁ = [tex]90*\frac{5280}{3600}[/tex] ( since 1 mi = 5280ft and 1 hour is equivalent to 3600 seconds)
V₁ = [tex]\frac{475200}{3600}[/tex]
V₁ = 132
V₂ = 45 mi/h
V₂ = [tex]45*\frac{5280}{3600}[/tex]
V₂ = [tex]\frac{237600}{3600}[/tex]
V₂ = 66
Substituting both data into the above equation, we have;
[tex]f_r_l[/tex] [tex]= 0.01(1+\frac{(\frac{132+66)}{2} }{147})[/tex]
[tex]f_r_l[/tex] [tex]= 0.01(1+\frac{(\frac{198)}{2} }{147})[/tex]
[tex]f_r_l[/tex][tex]= 0.01(1+\frac{99 }{147})[/tex]
[tex]f_r_l[/tex] [tex]= 0.01(\frac{147+99 }{147})[/tex]
[tex]f_r_l[/tex] [tex]= 0.01(\frac{246 }{147})[/tex]
[tex]f_r_l[/tex] [tex]= 0.016735[/tex]
Now, to calculate the breaking efficiency of the car [tex](n_b)[/tex]; we apply the formula:
[tex]S = \frac{Y_b(V_1^2-V_2^2)}{2g(n_b\beta+f_{rl}-sin\alpha g) }[/tex]
where;
the braking skid (S) = 410 ft
Value for braking mass factor for automobiles = 1.04
Value for Coefficient (β) of the road adhesion for both good and wet pavement = 0.90
Sin ∝ = 3% = 0.03
coefficient of the rolling friction [tex]f_r_l[/tex] = 0.016735
V₁ = 132 & V₂ = 66
Substituting our parameters in the above formula, we have;
[tex]410 = \frac{1.04 (132^2-66^2)}{2*32.2(n_b(0.9)+(0.016735)-0.03) }[/tex]
[tex]410 = \frac{1.04 (17424-4356)}{64.4(n_b(0.9)-(0.013265) }[/tex]
[tex]410 = \frac{1.04 (13068)}{57.96n_b-0.854266 }[/tex]
[tex]410 = \frac{13590.72}{57.96n_b-0.854266 }[/tex]
[tex]13590.72 = 410 (57.96_{n_b}-0.854266)[/tex]
[tex]13590.72 = 23763.6_{n_b}-350.24906[/tex]
[tex]13590.72+350.24906= 23763.6_{n_b}[/tex]
[tex]13940.96906=23763.6_{n_b}[/tex]
[tex]n_b=\frac{13940.96906}{23763.6}[/tex]
[tex]n_b=0.58665[/tex]
[tex]n_b[/tex] ≅ 0.587
[tex]n_b[/tex] = 58.7%
∴ The braking efficiency of the car = 58.7%
