1.A projectile is fired at speed $u=20\text{ m/s}$ at an angle $30^\circ$ above horizontal. Find maximum height, time of flight, and horizontal range.
Sol: $H = \frac{u^2 \sin^2 30^\circ}{2g} = \frac{400(0.25)}{19.6} = 5.1\text{ m}$. $T = \frac{2u\sin 30^\circ}{g} = \frac{40(0.5)}{9.8} = 2.04\text{ s}$. $R = \frac{u^2 \sin 60^\circ}{g} = \frac{400(0.866)}{9.8} = 35.3\text{ m}$.
2.A particle moves in a circle of radius $r=5\text{ m}$ at a constant speed $v=10\text{ m/s}$. Find centripetal acceleration.
Sol: $a_c = \frac{v^2}{r} = \frac{10^2}{5} = 20\text{ m/s}^2$ directed towards center.
3.Prove that the trajectory of a projectile is parabolic.
Sol: $x = (u\cos\theta) t \implies t = \frac{x}{u\cos\theta}$. $y = (u\sin\theta) t - \frac{1}{2}gt^2 = (\tan\theta) x - \frac{g}{2u^2\cos^2\theta} x^2$, which is of form $y = ax - bx^2$ (a parabola).
4.A swimmer can swim at $4\text{ km/h}$ in still water. River flows at $3\text{ km/h}$. In what direction should the swimmer head to cross shortest distance?
Sol: Shortest path is straight across: $\sin\theta = \frac{v_{river}}{v_{swimmer}} = \frac{3}{4} \implies \theta = \arcsin(0.75) \approx 48.6^\circ$ upstream from perpendicular.
