What is Projectile Motion?

Students appearing for JEE main advanced exam must prepare themselves thoroughly to secure a seat in any of the Indian Institutes of Technology (IITs). There is a vast amount of syllabus to be covered in the upcoming months. Students must sort out the subjects and topics by the marking scheme for each topic. All three subjects – Physics, Chemistry and Mathematics require an equal amount of time for preparation. However, it also depends on the extent of one’s preparation and their strong and weak areas.

Kinematics is one of the most important areas in Physics that a student must master thoroughly. The equations will come in handy throughout the entire year when they are studying physics, or even mathematics (vectors) for the matter of fact.

Below is a short and simple wrap-up of projectile motion in Physics, that carries a good weight in the JEE main advanced exam.

What is a Projectile?

A projectile is a body (object) that is thrown into the space which is in flight and only gravity acts on it.

What is Projectile Motion?

Projectile motion is a form of motion that is experienced by an object that makes it move along a curved path due to the action of gravity. It is motion where the only force acting upon the object (projectile) is gravity. Also, when an object is in flight after being thrown, it is said to be in projectile motion.

Example –A cricket ball or a javelin in the air.

What are the assumptions considered for Projectile Motion?

For projectile motion, the following assumptions are considered:

• The effect on the projectile is negligible due to the curvature of the Earth
• The impact on the projectile is insignificant due to the rotation of the Earth
• There is no resistance due to air

Projectile motion has two simultaneous independent rectilinear motions:

• Along x-axis (Responsible for the horizontal motion of the object; uniform velocity)
• Along y-axis (Responsible for the vertical motion of the object; uniform downward acceleration ‘g’)

Equations of Motion for Constant Acceleration

The most commonly used equations for motion of an object with constant acceleration are:

1. v = u – gt
2. s = ut – ½ gt²
3. v² = u² – 2 gs

Here:

• u = initial velocity
• g = acceleration due to gravity
• t = time
• s = displacement
• v = final velocity

Equations for Projectile Motion

If a projectile is launched with velocity ‘u’, it makes an angle ‘θ’ with x-axis. Once the projectile is in the air, the acceleration acting on it due to gravity is directed vertically downwards:

a= -g (combined)

a_x = 0                                                   A

a_y = -g                                                  B

Initial velocity ‘u’ is:

u_x = ucosθ                                        C

u_y = usinθ                                         D

x and y = 0, if initial position is the origin of the projectile

At any time ‘t’, displacement will be:

x = (u_x)t = (ucosθ)t

y = (usinθ)t – (½)gt²                           E

Velocity at time ‘t’can be obtained by:

u_x = ucosθ

u_y = usinθ– gt                                                F

The relation between ‘x’ and ‘y’ can be further explained by:

y= (tanθ)x – gx²/ (4u²cos²θ) G

Note: ‘x’ remains constant throughout the motion, and only ‘y’component changes.

Time of Flight

In equation G, the total time t_fduring the flight can be calculated by using y = 0. It means there is zero displacement in the vertical direction.

t_f = 2usinθ/g                                    H

t_f is the time of the flight of the concerned projectile.

t_f = 2t_m(t_m is the time taken to reach maximum height)

Maximum Height

h_max = (u²sin²θ)/2g

The maximum height h_mcan be calculated by substituting t = t_m

Range of the Projectile

Horizontal range (r) of the projectile is the distance covered by it from its initial position to the position where is passes y = 0.

Horizontal Range = Horizontal component of velocity (u_x) x Total Time of Flight (t)

It is the distance traveled during the flight t_f

r = ucosθ x (2usinθ)/g

i.e.

r = (u²sin(2θ))/g

The Equation of Trajectory (Why Projectile Motion is Parabolic)

Equations of motion tell us that:

a = t(ucosθ)                                    K

b = usinθ x t – ½ x t²/g             L

where (a,b) is the position of a body at time (t)

By taking Equation Kand L

b = (atanθ – ga²)/2u²cos²θ (Equation of Trajectory in Projectile Motion)

Projectile motion is a two-dimensional motion. It can also be said that it is a form of motion in a plane.