Projectile Motion

Class XI

Derivation of Projectile Equation

An object that is in flight after being thrown or project is called a projectile.

Consider about an object that is projected with an angle θ from horizon. The velocity in which the object is projected is v.

Velocity along vertical = vsinθ
Velocity along horizon = vcosθ
Time of flight of the projectile = T
Maximum Height of the projectile = H
Horizontal Range of the projecile = R
Time ascent = Time of decent = t
Time of flight = T = Time of ascent + Time of decent = t + t = 2t

Maximum distance covered by projectile along vertical direction = Y

Maximum distance covered by projectile along horizontal direction =X

Therefore,

Y = vsinθ.T - gT²/2  -------------------- (i)

X = vcosθ.T

T = X/vcosθ

Now putting the value of T

Y = vsinθ.X/vcosθ - gX²/2cos²θ

Υ = X.tanθ - gX²/2cos²θ ------------------------------ (ii)

Τhis is known as equation of Trajectory. This represent the equation of parabola.

Hence the path of the projectile is a parabola.

Maximum height of a projectile (H).

Here time of ascent = t = time of flight/2 = T/2 (Time taken by projectile to attains maximum height)

At maximum height the velocity of the projectile = 0

Now from kinematic equation

0 = vsinθ - gt

t = vsinθ /g

Time of ascent = Time of decent = t = vsinθ /g

Time of flight = T = 2vsinθ /g

Therefore, 

H = vsinθ.t - gt²/2

Now putting the value of t.

H = (vsinθ )² /g - g(vsinθ)²/2g²

H =  (vsinθ )² /2g 

This is the equation of maximum height.

Horizontal Range = R = vcosθ.T

Now putting the value of T

R = vcosθ.2vsinθ /g

R = v²sin2θ/g

Important equations derive from projectile are

1. Υ = X.tanθ - gX²/2cos²θ
2.  t = vsinθ /g
3. T = 2vsinθ /g
4. R = v²sin2θ/g