Time of flight: It is the time taken by the body from the instant it is realized till it strikes a position on the same horizontal plane as the point of projection.

To calculate time of flight, we can find the time when vertical displacement is zero i.e.

0 = u sinΘ t - 1/2gt²  (u sinΘ - 1/2gt) t = 0       t = 0     or         t = 2usinΘ/g

Thus ,  Time of flight T =2usinΘ/g

Range: During this time horizontal element of velocity has taken the particle through a distance x horizontally

Where,            x = u cosΘ * T = u cosΘ * (2usinΘ/g) = u²sin2Θ/g

This distance, we call as range on horizontal plane , R=u²sin2Θ/g   

            R_max = u²/g      for Θ = 45°

For a given velocity same range can be obtained for an angle Θ and angle (90° - Θ) i.e.

 R = u²sin2(90^o-Θ)/g = u²sin2(180^o-Θ)/g = u²sin2Θ/g     

To calculate maximum height, the vertical component of velocity equals to zero, when the particle is at the highest point from the ground. At that time particle has only horizontal component of velocity i.e. = u cosΘ

            0 = (u sinΘ)² - 2gHmax

            H_max= u²sin2Θ/2g

The motion of projectile can be analyzed through vector notation also for e.g. velocity of projectile (v) at any time t can be written as

Hence, the magnitude of velocity (speed) v at any time is given as ,

           (The path of a projectile is called its trajectory)

Trajectory of projectile

y = u sinΘ t - 1/2gt²                 . . . (1)      (vertical displacement at any time t)

x = u cosΘ t                             . . . (2)      (Horizontal displacement at any time t)

Using t =x/ucosΘ

We get, y=xtanΘ-gx²/2ucos²Θ Which suggest that trajectory of projectile is parabola

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