Motion in Two Dimensions

Projectile motion

Time, height, range
$T = \frac{2 u sin θ}{g},$ $H = \frac{u ^{2} sin ^{2} θ}{2 g},$ $R = \frac{u ^{2} sin 2 θ}{g}$
Maximum range
$R_{m a x} = \frac{u ^{2}}{g}$ at $θ = 4 5^{\circ};$ $θ$ and $9 0^{\circ} - θ$ give the same range
Trajectory
$y = x tan θ - \frac{g x ^{2}}{2 u ^{2} cos ^{2} θ} = x tan θ (1 - \frac{x}{R})$
$tan θ = \frac{4 H}{R}$
On an incline (angle α, thrown at θ to it)
$T = \frac{2 u sin θ}{g cos α},$ $R_{m a x} = \frac{u ^{2}}{g ( 1 + sin α )}$ up the incline

Relative velocity
$v_{A B} = v_{A} - v_{B}$
Shortest time
$t = \frac{d}{v _{b}}$ (head straight across) $,$ drift $= \frac{v _{r} d}{v _{b}}$
Shortest path
$sin θ = \frac{v _{r}}{v _{b}}$ (upstream) $,$ $t = \frac{d}{v _{b}^{2} - v _{r}^{2}}$
Rain–man
$v_{rain, man} = v_{rain} - v_{man}$
Hold the umbrella along the relative velocity of rain.