Question

Two banked curves have the same radius. Curve A is banked at 10.9 °, and curve B is banked at an angle of 18.8 °. A car can travel around curve A without relying on friction at a speed of 17.1 m/s. At what speed can this car travel around curve B without relying on friction?

Answer 1

if no friction is involved in the force balance, we know the horizontal component of the normal force must equal the centripetal force; this component has magnitude Nsin(theta)=mv^2/r

the vertical component of N must equal the weight of the car, or Ncos(theta)=mg

divide these two relations to get tan(theta) =v^2/rg or v^2= r g tan(theta)

now, we know that when theta=14 deg v=18m/s, so we have

tan10.9=17.1^2/(rg) =>r=154.94m

now, find v^2 when theta=18.8 deg

v^2=154.94m x 9.8m/s/s x tan18.8 => v=22.73m/s

Answer 2

we know,
tan(theta) = v^2(g*r)

tan(theta_A) = Va^2/(g*r) --(1)

tan(theta_B) = Vb^2/(g*r) --(2)

devide eqn 2 with eqn 1

tan(theta_B)/tan(theta_A) = (Vb/Va)^2

vb = va*sqrt( tan(theta_B)/tan(theta_A))

= 17.1*sqrt(tan(18.8)/tan(10.9))

= 22.74 m/s