Projectile Motion: Complete 2D Motion Guide for Exams

Projectile Motion is one of the most important chapters in physics for engineering entrance exams such as JEE Main, KEAM, MHT-CET, WBJEE, EAMCET and other state-level tests. This topic forms a strong foundation for understanding motion in two dimensions, vectors, real-life trajectories, and problem-solving involving components of velocity and acceleration. Even though the concept looks simple, exam questions are often wrapped with conceptual twists, angle-based formulas, and time-position relations.

In this blog, we will cover projectile motion in a complete, simplified, exam-oriented manner. If you understand this chapter deeply, many related chapters—like relative motion, 2D kinematics, and Newton’s laws—become easier.

What Is Projectile Motion?

Projectile motion is the motion of an object thrown into the air under the influence of gravity alone. Examples include a football kicked in the air, a stone thrown at an angle, a bullet fired from a gun, and even water from a fountain.

The key point is:
✔ Only gravity acts vertically downward
✔ There is no horizontal acceleration

This means the horizontal and vertical motions can be treated independently, but they are linked through time.

Understanding 2D Motion: Horizontal vs Vertical Components

When an object is projected with a velocity u making an angle θ with the horizontal, the motion splits into two components:

Horizontal Component

u_{x} = u \cos θ

There is no horizontal acceleration, so horizontal velocity remains constant.

Vertical Component

u_{y} = u \sin θ

Vertically, gravity acts downward with acceleration g, making vertical velocity change with time.

This “two-component” structure is what makes projectile motion a beautiful example of 2D physics.

Equations of Projectile Motion

Because horizontal and vertical motions behave differently, we write separate equations for both.

Horizontal Motion (Uniform Motion)

x = u \cos θ \cdot t

Vertical Motion (Uniformly Accelerated)

y = u \sin θ \cdot t - \frac{1}{2} g t^{2}

These two equations together describe the complete 2D trajectory of a projectile.

Time of Flight (T)

The total time a projectile stays in the air is called the time of flighe.

H = \frac{u^{2} \sin^{2} θ}{2 g​}

This is an extremely important result.
Observations:

Time of flight ∝ sinθ → larger angle → stays longer in air
At θ = 90°, time of flight is maximum
At θ = 0°, time of flight = 0

Maximum Height (H)

This is the highest vertical point reached by the projectile.

H = \frac{u^{2} \sin^{2} θ}{2 g​}

Key points:

Only affected by vertical velocity
Directly proportional to sin²θ
Maximum height occurs at θ = 90°

Horizontal Range (R)

Range is the total horizontal distance travelled before the projectile lands.

R = \frac{u^{2} \sin 2 θ}{g}

Important observations:

Depends on sin2θ, not sinθ
Maximum when 2θ = 90° → θ = 45°
Range for angles θ and (90° − θ) is the same
Example: Range at 30° = Range at 60°

This symmetry is useful in many exam problems.

Velocity at Any Time (t)

Horizontal velocity remains unchanged:

v_{x} = u \cos θ

Vertical velocity changes due to gravity:

v_{y} = u \sin θ - g t

Resultant velocity:

v = \sqrt{v_{x}^{2} + v_{y}^{2}}

Direction of velocity:

\tan ϕ = \frac{v_{y}}{v_{x}}

These relations help in problems where you must find angle of direction after some time.

Equation of the Trajectory

The path of a projectile is a parabola. Eliminating time (t) from x and y equations gives:

y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}

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

Time to Reach Maximum Height

t_{\text{max}} = \frac{u\sin\theta}{g}

At this point, vertical velocity becomes zero.

Projectile Dropped/Horizontally Launched from Height

If an object is projected horizontally from a height:

Horizontal velocity: u
Vertical velocity initially: 0
Time to hit ground
$t = \sqrt{\frac{2 h}{g}}$
Horizontal range:
$R = u \cdot t$

These problems occur frequently in exams because they combine free fall and horizontal motion.

Projectile From Height with Angle

When a projectile is launched from a height:

Time increases
Range increases
Landing angle changes
The symmetric properties no longer hold

Most entrance exam questions in advanced level deal with this scenario.

Relative Projectile Motion

This appears in tougher JEE problems. Examples include:

Two objects thrown simultaneously
One thrown upwards, another downwards
Two projectiles crossing each other
One projectile fired from a moving platform

Key idea:
Use relative velocity in 2D:

{\vec{v}}_{A B} = {\vec{v}}_{A} - {\vec{v}}_{B}

Most Repeated Concepts in Entrance Exams

Here are the most frequently tested concepts:

1. Maximum range occurs at 45°

This appears in at least one question every year.

2. Use of sin2θ identity

Especially in matching angle questions.

3. Horizontal velocity constant

Very common conceptual question.

4. Time of flight and maximum height calculations

Often asked as quick numericals.

5. Trajectory equation identification

Recognizing that the path is parabolic.

6. Projectiles from height

Popular in advanced exams like JEE.

7. Relative projectile problems

Test understanding of vectors and 2D motion.

Common Mistakes Students Make

Using vertical equations for horizontal motion
Forgetting to break initial velocity into components
Mixing degrees and radians in trigonometry
Using sinθ instead of sin2θ for range
Incorrectly determining sign of acceleration (a = –g upward)

Avoid these to score better in exams.

Conclusion

Projectile motion is a fundamental chapter that tests a student's ability to visualize and analyze 2D motion using vector components. It is not only essential for entrance exams but also forms the foundation for advanced mechanics topics. Mastery of this chapter enables students to solve a wide range of problems—from simple motion equations to challenging multi-projectile scenarios.

Understanding the symmetry, equations of motion, and geometric interpretation of the trajectory allows students to handle any exam question smoothly. With consistent practice, solving projectile motion problems becomes easy and enjoyable.