Projectile Motion (Kinematics)

 

 

Introduction to Projectile Motion

An object projected by an external force that continues to move by its own inertia is known as a projectile, and its motion is referred to as projectile motion. Common examples include a football kicked by a player, an arrow shot by an archer, water sprinkling from a fountain, or an artillery shell fired from a gun.

Conditions for Parabolic Trajectory

For a projectile to move on a parabolic trajectory under simplified conditions (neglecting wind and air resistance and assuming uniform gravity), the following must be fulfilled:

• Acceleration vector must be uniform.
• Velocity vector never coincides with the line of acceleration vector.

Projectile motion is one of the most important applications of two-dimensional motion in physics. It explains how objects move when they are thrown or projected into the air and then move only under the influence of gravity.

1.What is Projectile Motion?

When an object is thrown into the air with some initial velocity and then continues to move only under the influence of gravity, the motion is called projectile motion.

In this motion:

• The object moves freely under gravity
• Air resistance is usually neglected
• The path followed by the object is parabolic

Examples of Projectile Motion

Projectile motion appears in many everyday situations:

• A football kicked by a player
• An arrow shot by an archer
• Water coming out of a fountain
• A bullet fired from a gun
• A ball thrown into the air

All these objects move in curved trajectories, known as parabolic paths.

2.Shape of the Projectile Path

 

 

The path followed by a projectile is called its trajectory.

Because the horizontal and vertical motions combine together, the trajectory becomes a parabola.

Important points of the trajectory:

• Point of Projection – where the object is thrown
• Maximum Height – highest point reached
• Point of Landing – where the object hits the ground
• Range – horizontal distance travelled

3.Conditions for Projectile Motion

For motion to be considered projectile motion:

I. Acceleration must remain constant

The only acceleration acting is gravitational acceleration (g).

g = 9.8  m/s²

II.Velocity should not be in the same direction as acceleration

If velocity is purely vertical, motion becomes free fall, not a projectile.

IV.Analysis of Projectile Motion

Projectile motion can be understood as a combination of two independent motions:

Direction	Type of Motion
Horizontal	Uniform motion
Vertical	Uniformly accelerated motion

These motions are independent of each other.

V.Horizontal and Vertical Components

 

 

 

Suppose a projectile is thrown with velocity (u) at angle θ.

Velocity components:

Horizontal component

u_x​=ucosθ
Vertical component

u_y​=usinθ

VI.Vertical Motion Equations

Vertical motion behaves like uniformly accelerated motion.

Acceleration is downward due to gravity.

Velocity equation

v_y​=u_y​−gt

Displacement equation

y=u_y​t−1​/2gt²

Velocity–displacement relation

v_y^2​=u_y²​−2gy

VII.Horizontal Motion Equations

In horizontal direction:

• No acceleration
• Velocity remains constant

Therefore,

x = u_x t

where

u_x​=ucosθ

VIII.Equation of the Trajectory

The trajectory equation relates x and y coordinates of the projectile.

By eliminating time (t), we get:

y=x tanθ−g​ x²/2u²cos²θ

This is the equation of a parabola, proving the projectile path is parabolic.

IX.Time of Flight

The time of flight is the total time the projectile remains in the air.

At the highest point:

v_y​=0

Time of flight:

T=2u_y​​/ g

Since $u_{\mathrm{_y}}=\mathrm{<span\; class="mord">u</span><span\; class="mop">sin</span><span\; class="mord">\theta </span><span\; style="font-family:\; "Times\; New\; Roman",serif;\; font-size:\; 14pt;\; mso-fareast-font-family:\; "Times\; New\; Roman";\; mso-fareast-language:\; EN-IN;\; mso-font-kerning:\; 0pt;\; mso-ligatures:\; none;"></span>}$

T=2usinθ​/ g

X.Maximum Height

Maximum height occurs when vertical velocity becomes zero.

H=u_y²​​ /2g

Substituting u_y​=usinθ:

H=u²sin²θ/2g

XI.Horizontal Range

Horizontal range is the horizontal distance travelled by the projectile.

R=u_x​T

Substitute values:

R=2u_x​u_y​​ / g

Using trigonometric identity:

R=u²sin2θ​/g

 XII.Condition for Maximum Range

Range is maximum when:

θ=45^∘

Maximum range:

R_max​ =u^2​/g

XIII.Alternate Trajectory Equation

If range is known:

y=xtanθ(1−x​/R)

This form is useful in advanced projectile problems.

 Key Parameters of Projectile Motion

Term	Definition
Point of Projection	The point from where the object is projected.
Point of Landing/Target	The point where the object falls on the ground.
Velocity of Projection ($u$)	The velocity with which the object is thrown.
Angle of Projection ($\theta$)	The angle the velocity of projection makes with the horizontal.
Horizontal Range ($R$)	The distance between the point of projection and the point of landing.
Maximum Height ($H$)	The height from the ground of the highest point reached during flight.
Time of Flight ($T$)	The total duration for which the object remains in the air.

XIV.Key Summary Table

Quantity	Formula
Horizontal velocity	ucosθ
Vertical velocity	usinθ
Time of flight	T=2usinθ​/g
Maximum height	H=u2sin2θ​/2g
Range	R=u2sin2θ​/g
Maximum range	Rmax​ =u2​/g

⭐ Final Concept

Projectile motion is the combination of two independent motions:

• Uniform motion horizontally
• Uniformly accelerated motion vertically

Their combination produces the parabolic trajectory.

Understanding this concept helps solve many NEET and JEE problems in kinematics.