Maximum Height, Time of Flight and Range of a Projectile

The initial position of the projectile is at the origin O at t = 0. The projectile is launched with an initial velocity u at an angle θ, known as the angle of elevation, to the x-axis.

Its components in the x and y directions are:

ux = u cosθ

uy = u sinθ

Let ax and ay be the horizontal and vertical components, respectively, of the projectile’s acceleration.

ax = 0

ay = –g = –9.8 m s–2

Horizontal Motion

vx = ux (Since ax = 0)

x = uxt = u cosθ t

Vertical Motion

vy = uy – gt = u sinθ – gt

y = uyt – ½gt2 = u sinθ t – ½gt2

The horizontal motion is motion with constant velocity and the vertical motion is motion with constant (downward) acceleration.

Maximum Height

As the projectile travels through air, it climbs up to some maximum height (h) and then begins to come down. At the instant when the projectile is at the maximum height, the vertical component of its velocity is zero. This is the instant when the projectile stops to move upward and does not yet begin to move downward.

vy = 0

0 = uy – gt = u sinθ – gt

Time taken to rise taken to the maximum height is given by

t = (u sinθ)/g

At the maximum height h attained by the projectile, the vertical velocity is zero.

h = (u2 sin2θ)/2g

Time of Flight

The time of flight of a projectile is the time interval between the instant of its launch and the instant when it hits the ground.

T = 2t = (2u sinθ)/g

Range

Range is the distance traveled horizontally by the projectile. The range R of a projectile is calculated simply by multiplying its time of flight and horizontal velocity.

R = ux × T

R = (u cosθ)(2u sinθ)/g

R = (u2 sin2θ)/g

R will be maximum for any given speed when sin 2θ = 1 or 2θ = 90°. Thus, for R to be maximum, θ = 45°.