Introduction
(i) Any particle, which once thrown, moves freely in gravitational field of the earth, is defined as a projectile.
(ii) It is an example of two dimensional motion with constant acceleration.
(iii) Parabolic motion can be considered as two simultaneous motions in mutually perpendicular directions viz.
(a) Horizontal and (b) Vertical
1. Projectile Thrown From The Ground Level
(i) The particle is thrown from the ground level at an angle q from the horizontal velocity u.

(ii) Initial velocity can be resolved into two components
u cos θ = Horizontal component
u sin θ = Vertical component

From Horizontal Component we obtain	From Vertical Component we obtain
(a) Range (b) Velocity along horizontal direction after time t (c) Horizontal displacement after time t	(a) Maximum height (b) Time of flight (c) Time taken to reach maximum height (d) Vertical displacement after time t (e) Vertical velocity

Special Note :
The horizontal component of velocity (u cos θ) remains constant where as the vertical component changes constantly due to acceleration due to gravity 'g'.

In Horizontal Direction	In Vertical Direction
(a) Initial Velocity ux= u cos θ (b) Acceleration = 0 (c) Velocity after time t vx= u cos θ (d) Horizontal displacement after time t, x = ux.t = u cos θ t	(a) Initial Velocity uy = u sin θ (upward) (b) Acceleration a = g (downward) (c) Velocity after time t, vy = uy – gt = u sin θ – gt (d) Vertical displacement after time t, y = uyt – 1/2 gt2 = u sin θ t – (1/2) gt2 (e) Velocity at height h above the surface vy2 = uy2 – 2gh = u2 sin2 θ – 2gh vy = ± (one going upwards and other going downwards)

1.1 Velocity at a general point P(x, y) :


The direction of v from horizontal

1.2 Displacement :
= x+ y ⇒
In fig. OP gives the displacement of projectile in the position (x,y) tan b = y/x, b is the angle made by displacement with horizontal in the position (x, y)

1.3 Trajectory equation :
y = u sin θ t – (1/2) gt² and x = (u cos θ)t
From these equations, (eliminating t)

1.4 The maximum height reached by the projectile :

(u_y = vertical velocity)
1.5 Time of flight of the projectile :
Time taken to reach max. height

(using v = u + gt)
The time interval from initial launch of projectile upto its return to the ground level is known as the time of flight (T) of projectile

where u_y= vertical velocity
1.6 Horizontal Range of The Projectile :
The horizontal distance covered by the projectile during its time of flight is known as the horizontal range of the projectile.

(using v² = u²+ 2gh)
(i) For maximum range, θ = 45º, R_max= u²/g
In this situation, H_max =
(ii) The range of the projectile is same when its angle of projection from the horizontal is either q or (90º – θ).
(iii) Also
where u_x = u cos θ = horizontal velocity
u_y = u sin θ = vertical velocity
1.7 The velocity of the projectile at any instant of time t :
and its direction from horizontal
Note : The trajectory of the missiles, targeted at very long ranges is not a parabola rather than it is an ellipse.
1.8 Change in Momentum :

(i) Initial velocity = u cos θ + u sin θ
(ii) Final velocity = u cos θ– u sin θ
Change in velocity for complete motion
Δ ==– = – 2u sin θ
(iii) Change in momentum for complete motion Δ = –
= m (–) = m (– 2 u sin θ) = –2 m u sin θ
(iv) Velocity at the heighest point == u cos θ change in momentum at highest point
(m– m) = m [u cos θ– (u cos θ + u sin θ)] = – m u sin θ
(v) Kinetic energy at highest point = E₀cos² θ where E₀ = Initial K.E.
(vi) Magnitude of velocity at height 'h' , by energy conservation law
⇒ (1/2) mu² + 0 = (1/2) mv² + mgh
⇒
Special Note :
(a) The direction of acceleration of the projectile is different from its velocity direction.
(b) The acceleration of the projectile remains constant in magnitude and direction that is why the path of its motion is parabola.