Projectile Motion

Projectile Motion:


In this case angle of projection is < 90^o. Projectile motion is motion in two dimensions and uniform acceleration in one dimension. For the entire motion acceleration of a projectile

is ''g'' downwards. For a projectile, path or trajectery is parabolic. If a body is projected with a velocity u at an angle to the horizontal.



max

i) Horizontal component of velocity u_x = u cos

ii) Initial vertical component of velocity u_y = u sin

iii) Velocity of projection

iv) Angle of projection = tan-1

v)


Velocity after time ''t'' :

i) a_x= 0; a_y= -g

ii) horizontal component of velocity through out the motion is constant.

iii) Vertical component of velocity changes with time

iv) Horizontal component of velocity V_x= u cos

v) Vertical component of velocity V_y= u sin - gt

vi) Velocity of the particle

vii) direction of motion w.r.t. to horizontal = tan^-1


Position of the projectile after time 't' :

i) If x and y represent the horizontal and vertical displacements with respect to the point of projection 't' seconds after projection x = (u cos ) t y = (u sin ) t - gt²

ii) Equation of trajectory is



iii) A = tan B =

Range and

Maximum height attained

Hmax=

Maximum height attained is maximum when = 90^o) (body projected vertically up)

Time of ascent t_a=

Time of flight t_f= (level ground)

Range R =

If projected from level ground range is maximum when angle of projection = 45^o.

Rmax.= ; = 45^o (level ground) time of light



Complementary angles of projection :

i) For a given velocity of projection angles of projection are and 90 - .

Ex : 30^o, 60^o

ii) Range in equal for complementary angles of projection (u = constant) R = Constant for and 90 - .

iii)If h1and h2are the maximum heights attained for complementary angles of projection

h₁+ h₂=




Rmax= 2 (h₁+h₂)

iv) If t₁and t₂are the times of flight for complementary angles of projection R = gt₁t₂

If is the angle of projection tan =

When range is maximum H_max= R/4. If R = H_max = tan^-1(4) ~ 76^o



Horizontal Projectile Motion :

When a body is projected, horizontally from the top of a tower path of the body is parabola relative to ground. Path of the object relative to pilot is vertically down.

If a body is projected with a velocity u horizontally.

i) u_x = u

ii) Initial vertical component of velocity is zero

iii) a_x= 0

iv) a_y= g

v) Vertical component of velocity and the velocity of the projectile increases with time


vi) Velocity of the projectile is maximum when the body strikes the ground



Velocity after time t :

i) horizontal component Vx = u

ii) vertical component Vy = gt =

(Y = distance fallen)

iii) Net velocity

iv) direction of motion or angle made by velocity vector with the horizontal is



v) Velocity on reaching the ground is




Note:- If a body is projected horizontally and another is dropped from the same height, both the bodies will take same time to reach the ground.


Position after time t :

(i) Horizontal displacement after time t. x = ut

(ii) Distance fallen in time ''t'' y = gt²

(iii) Equation of path :