On a banked race track, the smallest circular path on which cars can move has a radius of 120 m, while the largest has a radius of 151 m, as the drawing illustrates. The height of the outer wall is 10.4 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.

smallest speed is 19.85 m/s

largest speed is 22.26 m/s

Explanation:

given data

radius r1 = 120 m

radius r2 = 151 m

height of wall h = 10.4 m

to find out

smallest and largest speed

solution

first we find the angle θ by banking condition

tanθ = v² / rg ... 1

and here tanθ = h / r2 - r1

tanθ = 10.4 / (151 - 120)

tanθ = 0.335

θ = 18.52°

so from equation 1

tanθ = v² / r1g

tan18.52 = v² / 120 (9.8)

v² = 393.94

v = 19.85

so smallest speed is 19.85 m/s

and

from equation 1

tanθ = v² / r2g

tan18.52 = v² / 151 (9.8)

v² = 495.70

v = 22.26

so largest speed is 22.26 m/s