Kinematics — Projectile & Circular Motion (Medium)

Quick Recap — Projectile & Circular Motion

• Projectile (launch speed $u$, angle $\theta$): time of flight $T=\dfrac{2u\sin\theta}{g}$, max height $H=\dfrac{u^2\sin^2\theta}{2g}$, range $R=\dfrac{u^2\sin2\theta}{g}$. Range is maximum at $45^\circ$ and equal for $\theta$ and $90^\circ-\theta$.
• At the top the velocity is horizontal ($u\cos\theta$); if $H=R$ then $\tan\theta=4$.
• Horizontal launch from height $h$: time $\sqrt{\dfrac{2h}{g}}$, horizontal range $u\sqrt{\dfrac{2h}{g}}$.
• Uniform circular motion: speed constant but velocity (direction) changes; centripetal acceleration $a=\dfrac{v^2}{r}=\omega^2 r$, always toward the centre.

Worked mini-example. For $u=50$ m/s at $37^\circ$ ($\sin37^\circ=0.6$, $g=10$): $T=\dfrac{2(50)(0.6)}{10}=6$ s, $H=\dfrac{50^2(0.36)}{20}=45$ m, $R=\dfrac{50^2\sin74^\circ}{10}\approx240$ m.