how to find flight time/distance and greatest hight of projectiles?

a is acceleration, v is velocity, s is position

vertically:

a = - g

v = u - gt (by integrating with respect to t and setting initial speed u as constant of integration)

s = ut -gt²/2 + s₀ (by integrating again and setting initial position s₀ as constant of integration, although object if usually projected from origin so s₀ = 0)

horizontally:

a = 0

v = u (constant velocity as acceleration is zero)

s = ut + s₀ (again s₀ usually 0)

so for an object projected from the origin at speed u at an angle θ from the horizontal, the initial speed in the x direction is ucos(θ) and usin(θ) in the y direction

thus s_x = utcos(θ) and s_y = utsin(θ) - gt²/2

flight time: this is the value of t when s_y returns to zero

0 = utsin(θ) - gt²/2

so either t = 0 (at launch) or usin(θ) - gt/2 = 0

=> flight time is t = 2usin(θ)/g

flight distance: this is the value of s_x when s_{y = 0}

when s_y = 0, t = 2usin(θ)/g 

so s_x = utcos(θ) = 2u²sin(θ)cos(θ)/g = u²sin(2θ)/g (as sin(2θ) = 2sin(θ)cos(θ))

=> flight distance is x = u²sin(2θ)/g

greatest hight:

this is when s_y is at a stationary point, ie. when ds_y/dt = 0. This is also when v_y = 0.

v_y = usin(θ) - gt = 0

=> t = usin(θ)/g

=> s_y = utsin(θ) - gt²/2 = u²sin²(θ)/g -  u²sin²(θ)/2g =  u²sin²(θ)/2g

=> greatest height is h = u²sin²(θ)/2g

:)