Parabolic Trajectory: Launch Angle, Vertex and Range

Grades 10-11
Interactive Parabolic Trajectory: Launch Angle, Vertex and Range graph — click to explore
Click the graph to explore it interactively →
Question: Parabolic trajectory
Parabolic trajectory is a classic topic where physics and geometry meet. When an object is launched and affected only by gravity (ignoring air resistance), its path of motion is mathematically a two-dimensional parabola.
We usually break this motion into two independent components:
  1. Horizontal motion: since no force acts horizontally, the object moves at constant velocity. \begin{align*} x &= v_0 \cos(\theta) t \end{align*}
  2. Vertical motion: under gravitational acceleration g, the object undergoes uniformly accelerated motion vertically. \begin{align*} y &= v_0 \sin(\theta) t - \frac{1}{2} g t^2 \end{align*}
Eliminating the time parameter t gives us the trajectory's function: \begin{align*} y &= x \tan(\theta) - \frac{g x^2}{2 v_0^2 \cos^2(\theta)} \end{align*}
In this model, we can identify the following key features:
  • Launch angle ( \theta): determines the shape of the trajectory. On level ground, a 45^\circ angle theoretically gives the maximum range.
  • Vertex: the maximum height the object reaches, where its vertical velocity is zero.
  • Range: the horizontal distance traveled when the object returns to its launch height.
  • Symmetry: ignoring air resistance, the rising and falling paths are perfectly symmetric.
In the interactive graph below:
  • Red curve: the object's actual flight path.
  • Blue vector: the direction of the initial launch velocity.
  • Green point (V): the highest point of the trajectory (the vertex).
  • Purple point (P): the landing point, i.e. where the range ends.
  • Adjust the sliders to see how the initial speed and launch angle dynamically affect the width and height of the whole parabola.
physicsprojectile-motionparabolas