Two-Dimensional Projectile Motion Mechanics and Calculations

Fundamentals of Two-Dimensional Projectile Motion

• Horizontal Launch Kinematics

  • When an object is launched purely horizontally from a given height $h$, the initial vertical velocity component is zero: $V_{i(\text{Vertical})} = 0\,m/s$
  • The horizontal velocity component remains constant throughout the entire motion as there is no horizontal acceleration.
  • Under constant vertical gravitational acceleration $g$, the vertical motion formulas are:
  • Final vertical velocity: $V_f = g \times t$
  • Height / vertical displacement: $h = \frac{1}{2} g t^2$
  • Fall time / time of descent: $t = \sqrt{\frac{2h}{g}}$

• Inclined Launch Vector Resolution

  • For an object projected at an angle $\theta$ above the horizontal with an initial velocity $V_i$:
  • Initial horizontal velocity component: $V_{i(\text{Horizontal})} = V_i \times \cos(\theta)$
  • Initial vertical velocity component: $V_{i(\text{Vertical})} = V_i \times \sin(\theta)$

• Kinematic Behavior Along the Trajectory

  • Horizontal Axis (X-Axis):
  • The horizontal speed $V_H$ remains constant throughout the entire motion from launch to landing: $V_H = \text{Constant}$
  • Horizontal acceleration is zero: $a_x = 0\,m/s^2$
  • Vertical Axis (Y-Axis):
  • Ascending Phase (Rising): The vertical component of speed decreases as the object travels upward due to gravity acting in opposition to motion ($g = -9.8\,m/s^2$).
  • Maximum Height Peak (Apex): At the highest point of trajectory ($\text{max } h$), the vertical speed reduces to zero: $V_{\text{Vertical}} = 0\,m/s$
  • Total Speed at Peak: At maximum height, the total speed of the projectile is equal purely to its horizontal speed component: $V_{\text{total}} = V_H = V_i \times \cos(\theta)$
  • Descending Phase (Falling): The vertical component of speed increases as the object falls back downward with gravitational acceleration ($g = +9.8\,m/s^2$).

Equations of Kinematics and Derived Projectile Formulas

• Fundamental Equations of Linear Vertical Motion

  • First equation of motion: $V_f = V_i + gt$
  • Second equation of motion: $h = V_i t + \frac{1}{2} g t^2$
  • Third equation of motion: $V_f^2 = V_i^2 + 2gh$

• Derived Formulas for Angled Projectile Motion

  • Time to Reach Maximum Height ($t$):
  • Represents the time duration for half of the journey (ascent phase).
  • Equation: $t = \frac{V_i \times \sin(\theta)}{g}$
  • Total Time of Flight ($T$):
  • Represents the full trajectory flight time duration from launch to landing.
  • Equation: $T = 2 \times t = \frac{2 \times V_i \times \sin(\theta)}{g}$
  • Maximum Height ($h$):
  • The maximum vertical displacement achieved above launch elevation.
  • Equation: $h = \frac{(V_i \times \sin(\theta))^2}{2g}$
  • Horizontal Range ($R$):
  • The total horizontal displacement covered during the full flight time.
  • Equation: $R = V_H \times T = (V_i \times \cos(\theta)) \times T$

Worked Example 1: Trajectory Analysis for $V_i = 19.6\,m/s$ at $\theta = 30^\circ$

• Given Parameters

  • Initial velocity: $V_i = 19.6\,m/s$
  • Launch angle: $\theta = 30^\circ$
  • Gravitational acceleration magnitude: $g = 9.8\,m/s^2$

• Step-by-Step Calculations

  1. Horizontal Speed Component ($V_H$):
  • $V_H = 19.6 \times \cos(30^\circ)$
  • $V_H = 16.9\,m/s$
  1. Initial Vertical Speed Component ($V_{i(\text{Vertical})}$):
  • $V_{i(\text{Vertical})} = 19.6 \times \sin(30^\circ)$
  • $V_{i(\text{Vertical})} = 9.8\,m/s$
  1. Time Required to Reach Maximum Height ($t$):
  • $t = \frac{V_i \times \sin(\theta)}{g} = \frac{19.6 \times \sin(30^\circ)}{9.8}$
  • $t = \frac{9.8}{9.8} = 1\,sec$
  1. Total Flight Time ($T$):
  • $T = 2 \times t = 2 \times 1 = 2\,sec$
  1. Maximum Horizontal Range ($R$):
  • $R = V_H \times T = 16.9 \times 2$
  • $R = 33.8\,m$
  1. Maximum Height ($h$):
  • $h = \frac{(V_i \times \sin(\theta))^2}{2g} = \frac{(19.6 \times \sin(30^\circ))^2}{2 \times 9.8}$
  • $h = \frac{(9.8)^2}{19.6} = 4.9\,m$

Worked Example 2: Trajectory Analysis for $V_i = 50\,m/s$ at $\theta = 30^\circ$

• Given Parameters

  • Initial velocity: $V_i = 50\,m/s$
  • Launch angle: $\theta = 30^\circ$
  • Gravitational acceleration magnitude: $g = 9.8\,m/s^2$

• Step-by-Step Calculations

  1. Horizontal Speed Component ($V_H$):
  • $V_H = V_i \times \cos(\theta) = 50 \times \cos(30^\circ)$
  • $V_H = 43.3\,m/s$
  1. Initial Vertical Speed Component ($V_{\text{vert}}$):
  • $V_{\text{vert}} = V_i \times \sin(\theta) = 50 \times \sin(30^\circ)$
  • $V_{\text{vert}} = 25\,m/s$
  1. Time Required to Reach Maximum Height ($t$):
  • $t = \frac{V_i \times \sin(\theta)}{g} = \frac{25}{9.8}$
  • $t = 2.55\,sec$
  1. Total Flight Time for Full Trajectory ($T$):
  • $T = 2 \times t = 2 \times 2.55$
  • $T = 5.1\,sec$
  1. Maximum Height ($h$):
  • $h = \frac{(V_i \times \sin(\theta))^2}{2g} = \frac{(25)^2}{2 \times 9.8}$
  • $h = \frac{625}{19.6} = 31.89\,m$
  1. Maximum Horizontal Range ($R$):
  • $R = V_H \times T = 43.3 \times 5.1$
  • $R = 220.83\,m$