Projectile Motion

What is a Projectile motion?

When a particle is thrown obliquely near the earth’s surface, it moves along a curved path under constant acceleration that is directed towards the center of the earth (we assume that the particle remains close to the surface of the earth). The path of such particle is called a projectile and the motion is called as projectile motion. Air resistance to the motion of the body is to be assumed absent in projectile motion.

In a Projectile Motion, there are two simultaneous independent rectilinear motions:

  1. Along x-axis: uniform velocity, responsible for the horizontal (forward) motion of the particle.
  2. Along y-axis: uniform acceleration, responsible for the vertical (downwards) motion of the particle.

Accelerations in the horizontal & vertical direction of a particle in projectile motion: When a particle is projected in the air with some speed, the only force acting on it during its time in the air is the acceleration due to gravity (g). This acceleration acts vertically downward. There is no acceleration in the horizontal direction which means that the velocity of the particle in the horizontal direction remains constant.

Let us consider a ball projected at an angle θ with respect to horizontal x-axis with the initial velocity u as shown below:

Projectile Motion

The point O is called the point of projection; θ is the angle of projection and OB = Horizontal Range or Simply Range. The total time taken by the particle from reaching O to B is called the time of flight.

For finding different parameters related to projectile motion, we can make use of different equations of motions:

Projectile Motion

Total Time of Flight: Resultant displacement (s) = 0 in Vertical direction. Therefore, by using the Equation of motion:

gt2 = 2(uyt – sy) [Here, uy = u sin θ and sy = 0]

i.e. gt2 = 2t × u sin θ

Therefore, the total time of flight (t):

Projectile Motion

Horizontal Range: Horizontal Range (OA) = Horizontal component of velocity (ux) × Total Flight Time (t)

R = u cos θ × 2u×sinθg

Therefore in a projectile motion the Horizontal Range is given by (R):

Projectile Motion

Maximum Height: It is the highest point of the trajectory (point A). When the ball is at point A, the vertical component of the velocity will be zero. i.e. 0 = (u sin θ)2 – 2g Hmax [s = Hmax , v = 0 and u = u sin θ]

Therefore in a projectile motion the Maximum Height is given by (Hmax):

Projectile Motion

The equation of Trajectory: Let, the position of the ball at any instant (t) be M (x, y). Now, from Equations of Motion:

x = t × u cos θ . . . . . . (1)

y = u sin θ × t – 12×t2g. . . . . . (2)

On substituting Equation (1) in Equation (2):

Projectile Motion

This is the Equation of Trajectory in a projectile motion, and it proves that the projectile motion is always parabolic in nature.

More About Projectile Motion: Projectile motion is a type of two-dimensional motion or motion in a plane. It is assumed that the only force acting on a projectile (the object experiencing projectile motion) is the force due to gravity.

But how can we relate this kind of motion with the real world? How are the concepts of projectile motion applicable to daily life? Let us see some real life examples of motion in two dimensions.

All of us know about basketball. To score a basket, the player jumps a little and throws the ball in the basket. The motion of the ball is in form of a projectile. Hence it is referred as projectile motion. What advantage does jumping gives to their chances of scoring a basket? Now apart from basketballs, if we throw a cricket ball, a stone in a river, a javelin throw, an angry bird, a football or a bullet, all these motions have one thing in common. They all show projectile motion. And that is, the moment they are released, there is only one force acting on them- the gravity. It pulls them downwards, thus giving all of them an equal impartial acceleration of.

It implies that if something is being thrown in the air, it can easily be predicted how long the projectile will be in the air and at what distance from the initial point it will hit the ground. If the air resistance is neglected, there would be no acceleration in the horizontal direction. This implies that as long as a body is thrown near the surface, the motion of the body can be considered as a two-dimensional motion, with acceleration only in one direction. But how can it be concluded that a body thrown in air follows a two-dimensional path? To understand this, let us assume a ball that is rolling as shown below:

Projectile Motion

Figure 1 Motion in one dimension

Now, if the ball is rolled along the path shown, what can we say about the dimension of motion? The most common answer would be that it has an x-component and a y-component, it is moving on a plane, so it must be an example of a motion in two dimensions. But it is not correct, as it can be noticed that there exists a line which can completely define the motion of the basketball. Thus, it is an example of motion in one dimension. Therefore, choice of axis does not alter the nature of the motion itself.


Figure 2 Motion in Plane

Now, if the ball is thrown at some angle as shown, the velocity of the ball has an x-component and component and also a z-component. So, does it mean that it is a three-dimensional motion? It can be seen here that a line cannot define such a motion, but a plane can. Therefore, for a body thrown at any angle, there exists a plane that entirely contains the motion of that body. Thus, it can be concluded that as long as a body is near the surface of the Earth and the air resistance can be neglected, then irrespective of the angle of projection, it will be a two-dimensional motion, no matter how the axes are chosen. If the axes here are rotated in such a way that, then and can completely define the motion of the ball as shown below:

Motion in Two Dimensions

Figure 3 Motion in two dimensions

Thus, it can be concluded that the minimum number of coordinates required to completely define the motion of a body determines the dimension of its motion.

Types of Projectiles

Kinetic Projectiles: A kinetic projectile is defined as a projectile that does not possess any kind of charge. Some of the common kinetic energy projectiles include round shots and rocks that are usually blunt projectiles.

Among projectiles that don’t contain explosives are those propelled from railguns, coilguns, and mass drivers, and also kinetic energy penetrators. These weapons work by achieving a high muzzle velocity for the most part up to and crash into their objectives, changing over their kinetic energy into damaging heat waves.

Sports Projectiles: In projectile motion, propelling force is applied on the ball to make it move. The stronger the applied force more will be the propelling force. For example, Bowling, Pitching.

Delivery Projectiles: Several projectiles comprises of a biological substance or some kind of explosive charge. Apart from explosive playboard, projectiles can be drafted to create some kind of damage for instance poisoning.

Wired Projectiles: Some projectiles consists of a cable that is connected to an equipment after launching it. For instance wire guided missile that range up to four thousand meters. A grappling hook in order to draw the launcher to the target or a whaling harpoon in order to tow towards the launcher.

Stay tuned with Byju’s to learn more about projectile motion and its applications.

Practise This Question

For a projectile, the ratio of maximum height reached to the square of flight time is (g = 10 ms2)