On a banked race track, the smallest circular path on which cars can move has a

Question

On a banked race track, the smallest circular path on which cars can move has a radius of 122 m, while the largest has a radius of 159 m, as the drawing illustrates. The height of the outer wall is 13.9 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.

Explanation / Answer

T => angle of banked track tan(T)=13.9/(159-122) N => Normal reaction force on car Equilibrium equation : N = mg cos(T) -mV^2/r sin(T) (V: speed of car) For car to just stay on track N=0 mg cos(T) = mV^2/R sin(T) V = SQRT(R*g*tan(T)) Vmax = sqrt(Rmax*g*tan(T)) = sqrt(159*9.81*tan(T)) = 24.2 m/s Vmin = sqrt(Rmin*g*tan(T)) = sqrt(122*9.81*tan(T)) = 21.2m/s

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