Expression for Time of Flight, Horizontal Range and Maximum height Time of flight: It is the total time for which the projectile remains in air. We know, at the end of time of flight, the projectile reaches the point on the ground. So, vertical height gained by the y = 0. From vertical equation of motion,

Expression  for Time of Flight, Horizontal Range and Maximum height

Time of flight: It is the total time for which the projectile remains in air. We know, at the end of time of flight, the projectile reaches the point on the ground. So, vertical height gained by the y = 0.
From vertical equation of motion, we have:

y=(usinθ)T–12gT2

Now,

0=(usinθ)T–12gT2

∴T=2usinθg…….(i)

Equation (i) gives the time of flight of the projectile for velocity of projection u at an angle θ.
Horizontal range: The total distance covered by the body in projectile is called horizontal range.
Horizontal range = Horizontal component of velocity × time of flight

or,R=ucosθ×2usinθg

or,R=u2sin2θg

Maximum horizontal range: The horizontal range will be maximum when sin2θ is maximum.

i.e.,sin2θor,2θor,θ=1=sin90o=90o=45o

Maximum Height: The vertical component of velocity at maximum height is 0. So,

V2y0H=(usinθ)2–2gH=u2sinθ−2gH=u2sin2θ2g

Two angle of projections for the same range:
The range R for velocity of projection u and angle of projection θ is:

R=u2sin2θg

If (90-θ) be the another angle of projection,  then:

R2∴R1=R2=u2sin2(90–θ)g=u2sin(180–2θ)g=u2sin2θg