Projectile Motion Formula

Projectile motion is the motion of a projected object in flight which is a result of two separate simultaneously occurring components of motions. One component of motion is without any acceleration along a horizontal direction and the other along the vertical direction with constant acceleration due to the force of gravity (considering air resistance as negligible). In the next section, we will list down the Projectile Motion Formula or equations. (also known as Projectile trajectory equation)

Projectile motion formula or equations

Projectile Motion formula | equations of projectile motion
Figure 1: Projectile motion formula or equations

The initial velocity component along X-axis = V0x = V0 cosθ   and  the initial velocity component along Y-axis = V0y = V0sinθ.
(Air resistance is taken as negligible). g is the acceleration due to gravity.

Initial Horizontal VelocityV0x = V0 cosθ [along X axis]
Horizontal velocityVX = V0x = V0 cosθ [ uniform horizontal velocity]
Initial Vertical velocityV0y = V0sinθ [along Y axis]
Vertical velocityVy = V0sinθ  – g t [ acceleration = -g]
Horizontal displacementx= V0x.t = (V0 cosθ). t
Vertical displacementy= (V0sinθ ).t  – (1/2) g t2
Motion Path equation:y = (tanθ) x –   (1/2) g . x2/(V0 cosθ)2 [Parabolic]
Time to reach max height:tmax= (V0sinθ )/g
Total time of flight for a projectileTtot = 2(V0sinθ )/g
Maximum height reached:Hmax = ( V0sinθ )2/(2 g)
Horizontal range of a projectile:R = (V0sin2θ )/ g
Maximum possible horizontal range:Rmax = V0/ g
Projectile Motion formula | equations of projectile motion

Projectile Motion Solved Example

A ball is thrown at a speed of 28 m/s in a direction 30 degrees above the horizontal. Calculate (a) the maximum height (b) the time taken by the ball to return to the same level (c) the distance from the thrower to the point where the ball returns to the same level.

Solution:
V0 = 28 m/s, θ = 30 degrees
(a) The maximum height = Hmax = ( V0sinθ )2/(2 g) = (28 Sin 30)^2 / (2×9.8) m = 10 m

(b) the time taken by the ball to return to the same level = Total time = Ttot = 2(V0sinθ )/g = (2x28xsin 30) /9.8 = 2.9 s

(c) the distance from the thrower to the point where the ball returns to the same level = Horizontal range of a projectile = R = (V0sin2θ )/ g = (28x28xsin 60)/9.8 =69 m

Projectile Motion Formula
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