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Projectile Motion

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Last updated date: 12th Jul 2024
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What is Projectile Motion - Definition & Formula

We all know from Newton’s Universal Laws of Gravity that any object on Earth is constantly influenced by gravity and pulled towards the center of the gravitational field, that is the center of the Earth. Gravity influences any object suspended in the air to fall downwards, as in the case of fruits from trees. Also, if you launch an object up into the air, it travels into the air due to its inertia of motion. But since the object is also under the influence of gravity, it also dips down in the ground. Due to the influence of gravity, that object takes a curve path forward towards the ground before it eventually stops after touching the ground. The object which is launched into the air is called projectile, and this kind of motion is called projectile motion.


When we throw an object up, due to the force of gravity it eventually comes down. An object which is in motion in the air with no other force but gravity influencing it is a projectile. We can define a projectile as an object which is projected into the air with only downward gravitational force influencing it and continues its motion by the virtue of its own inertia.


 Projectiles can roughly be classified into three kinds:


  •   A projectile which is left to free fall from a considerable height

  •   A projectile which is launched directly upwards

  •   A projectile upwards at an angle towards the horizontal


What Is Projectile Motion?

Projectile motion is the motion experienced by an object in the air only under the influence of gravity. A projectile, that is launched into the air near the surface of the Earth’s and moves along a curved path, or in other words a parabolic path, under the action of gravity, assuming the air resistance is negligible. Study of such motion is called ballistics. The only force acting upon a projectile is gravity, which imparts a downward acceleration to the projectile. Because of the object’s initial inertia, an external force is unnecessary to maintain the horizontal velocity of the object. It also requires considering other external forces like friction from aerodynamic drag or internal propulsion when the projectile motion is at a larger scale.


When you throw a particle in an oblique path near the surface of the earth, it follows a curved route with constant acceleration. In such a situation, the particle's path is known as a projectile, and its motion is known as the projectile motion.


In a projectile motion, two separate rectilinear motions occur at the same time:


A particle's horizontal and vertical projectile motions are accelerated in the following ways: When a particle is sent into the air at a certain speed, gravity's acceleration is the only force acting on it during that period (g). Along the x-axis is uniform velocity, which is responsible for the horizontal particle, also known as the forward motion.


Along the y-axis: uniform acceleration, which is responsible for the particle's vertical (downwards) motion. This downhill acceleration has a vertical component. There is no acceleration in the horizontal direction, implying that the particle's velocity in that direction remains constant.


Maximum Projectile Height

Let us know the maximum height of a projectile after grasping what a projectile is and how it moves. The highest vertical location along the object's flight determines its maximum height. The object's starting velocity determines the projectile's range.


If v is the beginning velocity, g is gravity's acceleration, and H is the most significant height in metres, then = the initial velocity's angle from the horizontal plane (radians or degrees).


The formula that has been derived  for calculating the maximum height of a projectile is:


H=\[\frac{usin\theta }{g}\]


Ballistics, the study of projectile motion:

The science of mechanics that deals with the flight, behaviour, and effects of projectiles are called ballistics. The word ballistics comes from the word ballista, which was a siege machine used for war during the late 18th century. Ballista was a weapon that is used to hurl boulders or spears against the enemy troops or walls. The boulder or spears used against the enemy were launched by the ballista, after which it would travel in a parabolic path and land among the enemy. The operators of the ballista have to be concerned about the angle to launch the projectile and the path they had to travel to gauge the landing position of the ballista. This is the reason the study of projectile motion is called ballistics. The path travelled by the projectile after it is launched is called the ballistic trajectory. The study of ballistics is important in modern warfare since it involves calculating the trajectories of modern warfare equipment like bullets, unguided bombs, rockets, and such other equipment. It also involves the designing of projectiles to attain desired acceleration and performance.


The elementary equations of ballistics only take into account the initial velocity of the projectile and an assumed constant gravitational acceleration. Practical applications of ballistics have to consider a variety of other factors like air resistance, crosswinds, target motion, the varying acceleration due to gravity, and in the case the target is significantly far away, even the rotation of the earth.


The weapon with the highest potential for destruction is named the ballistic missile since it involves delivering warheads to a target by a missile which follows a ballistic trajectory and falls on the target under the influence of gravity.


Some Important One-Liners to Remember About Projectile Motion

  1. The kinetic energy is , and the linear momentum is  m.u.cosፀ at the highest point.

  2. The horizontal displacement of the projectile after t seconds is

  3. The vertical displacement of the projectile after t seconds is

  4. is the equation for the projectile's path.

  5. A projectile's course is parabolic.

  6. The kinetic energy is at the lowest position.

  7. The linear momentum is equal to m.u at the lowest position.

  8. The projectile's acceleration is constant during the motion and operates vertically downwards, equal to g.

  9. The angular momentum is equal to where h is the height.

  10. In the case of angular projection, the angle between velocity and acceleration might range from 0 to 180 degrees.


Key components which are to be remembered while dealing with projectile motion:


(image will be uploaded soon)


In the given diagram


The key components which we should remember when calculating projectile motion and solving problems are


  • Initial launch angle, or angle of projection

  • The initial angle of projection varies from 0° to 90° And it heavily influences the motion            of a projectile. 

  • Initial velocity, u

If the path of the projectile, its launching point, and its target is plotted on a graph, initial velocity can be expressed as x and y components. The initial velocity of the projectile, in that case, can be given as


uxor uy = u sinፀ


or we can use the Pythagoras theorem to find the velocity


  • Time of flight, T

The time of flight is the time period from when the object is launched to the time it reaches the ground. The time period the projectile is in the air depends on the initial velocity and the angle of projection. It is given as 


T = \[\frac{2usin\theta }{g}\]


  • Acceleration, a

There is no acceleration in the horizontal direction in case of projectile motion. The only acceleration present is the downward acceleration, that is acceleration due to gravity, also known as free fall. It is given as


ax = 0 or ay = -g


The value of g is equal to 9.8 m/s


  • Horizontal velocity, ux, and Vertical velocity, uy

At any point in the projectile motion, the horizontal velocity remains constant. On the other hand, vertical velocity varies linearly. This is because acceleration is constant at 9.8 m/s


  • Displacement, d or position at time t

At time given by t, the displacement components in a graph plotted with the origin of the projectile as the origin, the displacement components are


X = u.t.cosፀ and y = u.t.sinፀ-gt²


  • Maximum height, H

The maximum height of the projectile is the highest height the projectile can reach

It is given by


H = \[\frac{u^2sin^2\theta }{2g}\]


Range, R


The range of a projectile is the distance between the launch point and the target in a straight line.


It is calculated by R = \[\frac{u^2sin2\theta }{g}\]

FAQs on Projectile Motion

1.What is the basis behind projectile motion theory?

A projectile is defined as anyone hurled with a certain initial velocity and then permitted to move solely by gravity without the aid of an engine or fuel. This is how projectile motion is explained.

2.What assumptions are applied when studying projectile motion?

We assume a few scenarios to investigate the motion of a projectile.

The following is a list of them.

  • Because air has no frictional resistance, we presume the external resistance to be 0 unless otherwise mentioned in the numerical.

  • The influence of the earth's rotation and curvature is small; hence it is not considered.

  • At all places of projectile motion, the acceleration due to gravity is constant in magnitude and direction.

3.What are the components that influence projectile motion, and how do they affect it?

  • The faster the body moves at the start, the further it travels.

  • The greatest range is obtained by elevating 45 degrees.

  • The problematic one is wind velocity, which considerably complicates ballistics. You can guess qualitatively what happens whether the wind is blowing in the opposite direction of the motion, in the same direction as the motion, or sideways on the motion.

  • The moment of inertia can also influence whether the projectile rotates or spins around the flight axis, but this is outside the scope of most courses.

4.Give two instances where you can visualize projectile motion in real life?

Shooting a Canon

A cannonball fired does not travel in a straight line; instead, it travels on a curved route. This is because the ball is fired at an angle that causes it to move vertically and horizontally. As a result, the projectile motion is said to exist.


Sneezing

Sneezing is our body's natural reflex to eject foreign material from our mouth and nose. The particles and droplets that come out of your mouth travel in a projectile manner as they land on nearby objects and surfaces when you sneeze.

5.Is projectile motion demonstrated by cars turning on a curved road?

No, because there is no vertical component to a vehicle's turning, it cannot be classified as a projectile action. However, if the car is launched into the air while turning, it can be termed a projectile motion. When stunts are conducted, this is the situation. When an object moves in a projectile motion, it uses both horizontal and vertical velocity. One of the most important conditions for a stuntman to make a car jump along a ramp and land safely on the other side of the setup is vertical velocity.