Projectile Motion

For One-Dimensional Kinematics, the motion of falling objects is a simple one-dimensional kind of projectile motion with no horizontal movement.

The most significant thing to understand here is that movements along perpendicular axes are independent of one another and may therefore be evaluated individually. Because In this article, we will look at two-dimensional projectile motion, such as that of a football or other object with negligible air resistance.

Where both vertical and horizontal motion observed to be independent. For understanding two-dimensional projectile motion is to divide it into two motions, one horizontal and one vertical.

What is Projectile Motion?

When a particle is thrown obliquely near the earth’s surface, it travels along a curved path with constant acceleration toward the earth’s centre (we assume that the particle remains close to the surface of the earth). Such a particle’s route is referred to as a projectile path, and its motion is referred to as projectile motion.

As explained in Problem-Solving Basics for One-Dimensional Kinematics, falling object motion is a basic one-dimensional kind of projectile motion with no horizontal movement. In this part, we’ll deal with two-dimensional projectile motion, such as that of a football or other item with little to no air resistance.

The most essential thing to remember is that motion along perpendicular axes are independent of one another and hence could be evaluated individually. This was stated in Kinematics in Two Dimensions: An Introduction, where vertical and horizontal motions are independent. To explain two-dimensional projectile motion, break it into two motions: horizontal and vertical.

Important Terms

Point of Projection – The point from which the body is projected in the air is referred to as the point of projection.

Velocity of Projection – The velocity at which the body is hurled is referred to as the velocity of projection.

Angle of Projection – The angle of projection is the angle at which the body is projected with regard to the horizontal.

Horizontal Range – The range of the projectile is the horizontal distance travelled by the body while accomplishing projectile motion.

Projectile Trajectory – The route travelled by a projectile in the air is referred to as the projectile’s trajectory.

Derivation

The projectile motion is always in the shape of a parabola, which is denoted as, y = ax + bx²

(To simplify the calculation, projectile motion is computed without taking into air resistance in account.)

If a particle is projected obliquely near the earth’s surface, it travels in both horizontal and vertical directions at the same time. The movement of such a particle is referred to as Projectile Motion. Where a particle is thrown at an angle θ with an initial velocity of u.

When we throw an object with velocity u​, at angle θ with horizontal x-axis.

Horizontal component of velocity = u cos θ

Vertical component of velocity = u sin θ

At time t = 0

Displacement towards X axis Sx = 0

Displacement towards Y axis Sy = 0

At time T=t,

Displacement X axis Sx = 0 —– (1)

Displacement Y-axis Sy  = Uy t – 1/2(gt²)  −−−−−−−(2)

Now we will compute the following:

Time of Flight

Total displacement along the Y axis is (Sy) = 0.

Therefore after considering motion in Y direction, Sy  = Uy t – 1/2(gt²)

[Here, uy = u sin θ and Sy = 0]

0 = u sin θ – ½ (gt²)

t = 2usinθ/g

Therefore total time of flight is 2usinθ/g

Horizontal distance travelled (Hx)

Horizontal range can be given by

Hx = Horizontal component of velocity(u_x) * Total time(t)      [Here, u_x = u cosθ and t = 2usinθ/g]

Now,

H_x= ucosθ * 2usinθ/g

As a result, the Horizontal Range of the projectile is given by (H_x) =  2usin2θ/g

Since,   [sin2θ = 2cosθsinθ]

Maximum Height reached (H_max)

When the object reaches, the vertical component of the velocity (Vy) will be zero.

0 = 2(usinθ)  – 2gH_max                         [ Here, S = H_max , vy = 0 and uy = u sin θ ]

Therefore, Maximum Height reached (H_max ) of the projectile is given by

H_max = 2usin2θ/2g

Therefore the maximum height reached by an object is = 2usin2θ/2g

Application of Projectile Motion

1. In sports: Because most sports require the motion of a projectile, projectile motion is highly prevalent in sports (usually a ball). We can use physics to find the best angle of flight for a ball in order to maximise speed or distance.
2. In Programming and Animation: Another area where projectile motion is used is in programming. The problem for current programmers and animators is to correctly replicate real-life physics, whether for a television show or a computer game. Great video games try to replicate the physics as closely as possible, whether it’s the mechanics of a baseball being hit or the physics of a human falling from a given height.

Conclusion

From real life sports such as football, volleyball or cricket everywhere we can see the projectile motion and trajectory path. And not only in real life in virtual worlds like video games and animation projectiles matter a lot if any object is thrown. So knowledge of projectiles helps us to calculate the maximum height reached by the object, horizontal path followed by the object while in projectile motion. Although in this article we have covered the concept of projectile and the terms related to it. Hope this article will be beneficial to you.