Question

A circular racetrack of radius $300m$ is banked at an angle of ${15}^o.$ If the coefficient of friction between the wheels of a race car and the road is $\mathrm{0.2.}$
(a) What is the optimum speed of the race car to avoid wear and tear on its tires?
(b) What is the maximum permissible speed to avoid slipping?

Answer 1

(a) Optimum speed of car is given as

$tan\theta =\frac{v^2}{rg}$

$v^2=rgtan\theta$

Now putting all the values we get

$v^2=10\times 300\times tan15$

$Or,v^2=780$

Or, v = 28.35 m/s

(b) Maximum permissible speed to avoid slipping is given as

$v_m=\sqrt{Rg\left(\frac{\mu +tan\theta }{1-\mu tan\theta }\right)}$

Now putting all the values

$v_m=\sqrt{300\times 10\left(\frac{0.2+tan15}{1-0.2tan15}\right)}$

$Or,v_m=\sqrt{1458.7}$

$Or,v_m=38.1$ m/s

Hence, maximum permissible speed is 38.1 m/s