Projectile motion is a fundamental concept in physics, crucial for understanding the trajectory of objects under the influence of gravity. It's a cornerstone for many competitive exams, including JEE Physics. This module breaks down the core principles of horizontal and vertical motion, which are independent yet work together to define a projectile's path.

The key to analyzing projectile motion lies in recognizing that the horizontal (x) and vertical (y) components of motion are independent of each other. This means that changes in one direction do not affect the motion in the other, assuming no air resistance.

In the absence of air resistance, there are no horizontal forces acting on a projectile. According to Newton's first law, an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force. Therefore, the horizontal velocity ($v_x$) of a projectile remains constant throughout its flight. The horizontal displacement ($\Delta x$) can be calculated using the formula: $\Delta x = v_x \times t$, where $t$ is the time of flight.

The vertical motion of a projectile is governed by gravity. Gravity acts downwards, causing a constant downward acceleration, denoted by $g$ (approximately $9.8 \, m/s^2$). This means the vertical velocity ($v_y$) changes over time. We can use the standard kinematic equations to describe the vertical displacement ($\Delta y$), vertical velocity ($v_y$), and time ($t$): $v_y = v_{y0} + a_y t$ and $\Delta y = v_{y0} t + \frac{1}{2} a_y t^2$, where $v_{y0}$ is the initial vertical velocity and $a_y = -g$ (taking upward as positive).

By combining the independent horizontal and vertical motions, we can describe the complete trajectory of a projectile. The horizontal motion determines the range, while the vertical motion determines the maximum height and the time of flight. Understanding these components is essential for solving problems related to projectile motion in competitive exams.