1.What is the path (trajectory) of a projectile launched at an angle to horizontal?
Sol: Parabola.
2.At what angle of projection is the horizontal range of a projectile maximum?
Sol: $45^\circ$.
3.Define centripetal acceleration and write its formula.
Sol: $a_c = \frac{v^2}{r} = \omega^2 r$, directed towards the center of circle.
4.State the time of flight formula for a projectile.
Sol: $T = \frac{2u \sin \theta}{g}$.
5.A projectile is launched with velocity $20 \text{ m/s}$ at $30^\circ$ to horizontal. Find max height and range ($g=10\text{ m/s}^2$).
Sol: $H = \frac{u^2 \sin^2 \theta}{2g} = \frac{400 \times (1/4)}{20} = 5 \text{ m}$. $R = \frac{u^2 \sin 2\theta}{g} = \frac{400 \sin 60^\circ}{10} = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \approx 34.64 \text{ m}$.
6.A body moves in a circle of radius $5 \text{ m}$ with constant speed $10 \text{ m/s}$. Find centripetal acceleration.
Sol: $a_c = \frac{v^2}{r} = \frac{100}{5} = 20 \text{ m/s}^2$ towards center.
7.A river is flowing at $3 \text{ km/h}$. A swimmer can swim at $4 \text{ km/h}$ in still water. Find resultant velocity if he swims perpendicular to flow.
Sol: $v = \sqrt{3^2 + 4^2} = 5 \text{ km/h}$ at angle $\tan \theta = 3/4 \implies \theta = 37^\circ$ to bank direction.
8.Show that projection angles $\theta$ and $(90^\circ - \theta)$ yield equal horizontal ranges.
Sol: $R(\theta) = \frac{u^2 \sin 2\theta}{g}$. For $90^\circ - \theta$, $R = \frac{u^2 \sin(180^\circ - 2\theta)}{g} = \frac{u^2 \sin 2\theta}{g}$. Both ranges are equal!
