Derivation of Equation of Trajectory Explained for Students

The derivation of the equation of trajectory is a fundamental part of projectile motion in physics. This equation describes the path that a projectile follows under the action of gravity, providing the mathematical foundation for analyzing and predicting the motion of objects projected into the air.

Definition of Trajectory and Key Variables

The trajectory of a projectile refers to the curved path followed by an object under the influence of uniform gravity, neglecting air resistance. The standard derivation uses variables such as initial velocity ($u$), angle of projection ($\theta$), and gravitational acceleration ($g$).

A clear understanding of each variable is essential for solving projectile motion problems and avoids common confusion regarding direction and unit usage.

Stepwise Derivation of the Equation of Trajectory

Consider a particle projected with initial velocity $u$ at an angle $\theta$ with respect to the horizontal axis. Gravity acts downwards with acceleration $g$. The motion can be resolved into two components: horizontal (x-axis) and vertical (y-axis).

The horizontal velocity component is $u\cos\theta$, which remains constant throughout the flight because no horizontal force acts (ignoring air resistance). The corresponding displacement after time $t$ is:

$x = u\cos\theta \cdot t$

The vertical velocity component $u\sin\theta$ is affected by gravity. The vertical displacement after time $t$ is:

$y = u\sin\theta \cdot t - \dfrac{1}{2}gt^2$

Eliminate $t$ from these equations to express $y$ as a function of $x$:

$t = \dfrac{x}{u\cos\theta}$

Substitute this value of $t$ into the equation for $y$:

$y = u\sin\theta \left(\dfrac{x}{u\cos\theta}\right) - \dfrac{1}{2}g\left(\dfrac{x}{u\cos\theta}\right)^2$

$y = x\tan\theta - \dfrac{g x^2}{2u^2 \cos^2\theta}$

This is the general equation of the trajectory for a projectile launched from the origin over a horizontal surface. It mathematically represents the path traced during the motion.

For further details on the vector breakdown of motion components, refer to the Kinematics Overview.

Parabolic Character of the Trajectory Equation

The derived equation $y = x\tan\theta - \dfrac{g x^2}{2u^2 \cos^2\theta}$ contains a linear term ($x\tan\theta$) and a quadratic term in $x$ ($-\dfrac{g x^2}{2u^2 \cos^2\theta}$). The absence of a $y^2$ term confirms that the path is a parabola.

A trajectory is classified as parabolic if it is described by an equation where one variable is of degree two and the other of degree one. Here, the $x^2$ term ensures the path is always a parabola if gravity is constant.

The parabolic nature holds true in the absence of air resistance and for constant $g$. For more on projectile types, see Projectile Motion Explained.

Special Cases in the Equation of Trajectory

Certain particular cases simplify the trajectory equation. For horizontal projection ($\theta = 0^\circ$), the equation reduces to $y = -\dfrac{g x^2}{2u^2}$ since $tan\,0^\circ = 0$ and $cos\,0^\circ = 1$.

For vertical projection ($\theta = 90^\circ$), $cos\,90^\circ = 0$, so the range is zero and the projectile rises and falls along a straight line.

If projected from a height $h$, the formula modifies to $y = h + x\tan\theta - \dfrac{g x^2}{2u^2 \cos^2\theta}$. This adjusted equation frequently appears in advanced problems and examinations.

Summary of Steps for JEE Main and Class 11 Examinations

• Resolve initial velocity into $u\cos\theta$ and $u\sin\theta$
• Write horizontal and vertical displacement equations
• Eliminate $t$ to relate $x$ and $y$ directly
• Simplify the resulting formula to standard parabolic form
• Always use SI units in calculations

Accurate computation of each step is necessary when solving projectile motion numericals. Always verify angle units (radian for calculations, degrees for diagrams) and sign convention for gravity (downwards negative $y$ direction).

For deeper conceptual understanding, students may consult Motion In One Dimension, focusing on kinematic equations derivation.

Applications and Relevance of the Trajectory Equation

The derived trajectory equation is used extensively in JEE Main, NEET, and board-level examinations, helping to calculate projectile height, range, time of flight, or spatial coordinates.

Real-world applications include sports ballistics, engineering, forensics, and any system involving objects projected under gravity. The parabolic path is also essential for graphical analysis of motion and theoretical calculations.

Practicing trajectory-based numericals strengthens understanding and problem-solving skills, which are core requirements for JEE aspirants. Related concepts can be explored in Velocity Of Object And Image.

Key Points for Exam Preparation

• Memorize the standard trajectory equation
• Know unit conversions and standard sign conventions
• Practice by solving for angle, range, or coordinate using provided values
• Identify the nature of the curve as a parabola for constant $g$
• Recognize formula modifications for horizontal or elevated projections

Understanding the mathematics behind trajectory curves enables quick option elimination in competitive exams. For equations involving mass or force, refer to De Broglie Equation as an example of formula application.