# A roller coaster pictured below starts at point $A$ and speeds down a curved track with speed $u$. Assume that the friction between the roller coaster and the track is negligible even though it always stays in contact with the track. What is the speed of the roller coaster at point $B$?

(A) $\large\sqrt ( \normalsize u^2 + \large\frac{1}{3} \normalsize gh) \quad$(B) $\large\sqrt ( \normalsize u^2 + \large\frac{2}{3} \normalsize gh) \quad$ (C) $\large\sqrt ( \normalsize u^2 + \normalsize gh) \quad$(D) $\large\sqrt ( \normalsize u^2 - \large\frac{1}{3} \normalsize gh)$

Total energy at point $A =$ Kinetic + Potential energy $= \large\frac{1}{2}$$mu^2+mgh If u_b is the speed at point B, total energy at B = = \large\frac{1}{2}$$mu_b^2+mg (\large\frac{2}{3}$$h) From the law of conservation of energy, Total energy at point A = Total energy at point B, \Rightarrow \large\frac{1}{2}$$mu^2+mgh = \large\frac{1}{2}$$mu_b^2+mg (\large\frac{2}{3}$$h)$
Solving for $u_b \rightarrow u_b = \large\sqrt { \normalsize u^2 + \large\frac{2}{3} \normalsize gh}$
edited Jun 13, 2014