# Horizontal Range in Projectile Motion

## Horizontal Range in Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subjected to only the acceleration of gravity. The object is called a projectile and its path is called as trajectory.

What is Horizontal Range?

Horizontal Range is the distance covered by the projectile during its time of flight. Since there is no acceleration in horizontal direction (or) it is defined as the maximum distance covered in horizontal distance. The horizontal range depends on the initial velocity (u), the launch angle (θ) and the acceleration due to gravity (g).

Horizontal Range = Horizontal Velocity x Time of Flight

$$Horizontal\,\,Range\left( R \right)=u\,cos\theta \times \frac{2\,u\sin \theta }{g}=\frac{{{u}^{2}}\sin \theta \cos \theta }{g}=\frac{{{u}^{2}}\sin 2\theta }{g}$$.

∴ $$Horizontal\,\,Range(R)=\frac{{{u}^{2}}\sin 2\theta }{g}$$.

Where,

u = Initial Velocity (m/ sec),

R = Horizontal Range (m),

g = Acceleration due to gravity (m/ sec²),

θ = Angle of the initial velocity from the horizontal plane (degrees)

For a given initial velocity (u), horizontal distance depends on angle of projection. Since sinθ has maximum value of 1, the horizontal range will be maximum when sin 2θ = 1.

∴ sin 2θ = 1 = sin 90°

2θ = 90°

θ = 45°

So, to achieve the maximum horizontal range, the object must be projected at an angle of 45° with the ground.