Learning Outcomes

Upon completing this lesson, you will be able to:

• Define projectile motion and the key assumptions made in its analysis.
• Decompose the initial velocity of a projectile into its horizontal and vertical components.
• Analyze the horizontal and vertical components of projectile motion as two independent rectilinear motions.
• State and apply the kinematic equations for the horizontal (constant velocity) and vertical (constant acceleration) motion.
• Derive and use the trajectory equation to describe the parabolic path of a projectile.
• Calculate key parameters of projectile motion, such as maximum height, time of flight, and range.

Topics Covered

• The Core Concept of Projectile Motion
• Analysis of Horizontal and Vertical Motion
• The Kinematic Equations for Projectile Motion
• The Trajectory Equation
• Key Analysis Points: Maximum Height and Range

I. The Core Concept of Projectile Motion

Projectile motion is the motion of an object that has been thrown or projected into the air, subject only to the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.

The fundamental principle for analyzing projectile motion is that the complex, curved motion can be broken down into two simpler, independent one-dimensional motions:

1. A horizontal motion with zero acceleration.
2. A vertical motion with constant downward acceleration due to gravity.

The only variable that connects these two independent motions is time $(t)$. The time it takes for the projectile to travel horizontally is the same time it is moving vertically.

Key Assumptions: For our analysis, we will make two key assumptions to simplify the problem:

1. Air resistance is negligible. This is the most significant simplification. In reality, air drag affects the motion, but for many introductory problems, its effect is ignored.
2. The acceleration due to gravity $(g)$ is constant and directed vertically downward. We will use $g \approx 9.81 , m/s^2$.

II. Analysis of Horizontal and Vertical Motion

To begin any projectile motion problem, the first step is to establish a coordinate system (with the positive $y$-axis pointing upward) and resolve the initial velocity, $v_0$, into its horizontal and vertical components, $(v_x)_0$ and $(v_y)_0$, using the launch angle $\theta$.

$$ (v_x)_0 = v_0 \cos(\theta) $$

$$ (v_y)_0 = v_0 \sin(\theta) $$

A. Horizontal Motion ($x$-direction)

• Force: Since we neglect air resistance, there are no forces acting in the horizontal direction.

• Acceleration: According to Newton's Second Law $(F=ma)$, if there is no horizontal force, the horizontal acceleration is zero.

  $$ a_x = 0 $$

• Velocity: Since the acceleration is zero, the horizontal velocity never changes. It remains constant throughout the entire flight.

  $$ v_x = (v_x)_0 = v_0 \cos(\theta) $$

B. Horizontal Motion ($x$-direction)

• Force: The only force acting on the projectile is the force of gravity (its weight), which acts downward.

• Acceleration: The vertical acceleration is therefore constant and directed downward.

  $$ a_y = -g = -9.81 , m/s^2 $$

• Velocity: Since the acceleration is constant, the vertical velocity changes linearly with time. It decreases as the projectile rises, becomes momentarily zero at the peak, and increases in the negative direction as the projectile falls.

III. The Kinematic Equations for Projectile Motion

By applying the principles from the previous section, we can write a set of kinematic equations tailored specifically for projectile motion.

Horizontal Motion $(a_x = 0)$ Since the velocity is constant, there is only one equation needed to describe the horizontal position: