Conditional:

Elastic collision with wall

If a wall is in the path of a projectile at a distance x from the starting point and the collision is elastic then:

$t_{f} and H remain the same, distance covered after the collision will be in opposite x direction and same y direction as distance covered before collision$
$If x \geq \frac{R}{2} landing spot is at distance 2 x - R from the launch point in the same x direction as initial velocity$
$If x \leq \frac{R}{2} landing spot is at distance R - 2 x from the launch point in the opposite x direction to initial velocity$

$The angle of inclination is α$
$The acute angle of projectile be θ$
$t_{f} = \frac{2 u \sin θ}{g \cos α}$
$R = \frac{u^{2} [\sin (2 θ + α) - \sin α]}{g \cos^{2} α}$
$R is max when θ = \frac{π}{4} - \frac{α}{2} and, R_{\max} = \frac{u^{2}}{g (1 + \sin α)}$
$Max distance from inclined plane = \frac{u^{2} \sin^{2} θ}{2 g \cos α}$

For down the plane replace α with -α to get the corrected expressions

$V_{man} = V_{man|river} + V_{river}$
$Time taken to cross river, t = \frac{d}{v \sin θ}, θ = angle between V_{man|river} and bank of the river$ $v = | V_{man|river} |$
$Drift = (v \cos θ + u) * t {u = | V_{river} |}$
$For drift to be zero, \cos θ = \frac{- u}{v}, t = \frac{d}{\sqrt{v^{2} - u^{2}}}$
$If v < u minimum non-zero drift is when, \cos θ = \frac{- v}{u}$
$For swimming in a specific direction, Angle of V_{man|river} with direction of swim be α v \sin α = u \sin θ, θ is angle of direction of swim with river bank$

Initial distance = L Angle made between V_{rel} and initial displacement = θ Angles made by the moving bodies be α and β and, velocities be V_{a}, V_{b}

$Minimum distance = L \sin θ$
$| V_{rel} | = \sqrt{{| V_{a} |}^{2} + {| V_{b} |}^{2} + 2 | V_{a} | | V_{b} | \cos (α + β)}$
$Time to reach minimum separation is \frac{L \cos θ}{| V_{rel} |}$

N = m (g + a)

N = m (g - a)

Lose contact with floor

Will hit ceiling if enough time and N = m (a - g)

Will free fall

Force felt by binding object = m (a - g)

Only bodies with external force will move, use standard kinetics and a = \frac{F}{m}