Kinematics
Advanced
projectile
Source: High School Physics Olympiad
Problem Sets:
kinetics - olympiad
Problem
Several frictionless inclined planes share a common base but have different inclination angles. An object starts from rest at the top of each incline and slides freely down.
For what inclination angle $\theta$ is the time to slide to the bottom a minimum?
$\theta = 45°$
Let the common base have length $b$. For inclination angle $\theta$, the incline length is $L = \frac{b}{\cos\theta}$ and the acceleration along the incline is $a = g\sin\theta$.
From $L = \frac{1}{2}at^2$:
$$t^2 = \frac{2L}{a} = \frac{2b}{g\sin\theta\cos\theta} = \frac{4b}{g\sin 2\theta}$$The time is minimum when $\sin 2\theta$ is maximum, i.e. $\sin 2\theta = 1$, so $2\theta = 90°$ and $\theta = 45°$.