About the Projectile Motion Simulator

This interactive web tool is designed to help students, teachers, engineers, and physics enthusiasts visualize and understand the fundamental principles of projectile motion. By manipulating key physical parameters like initial velocity, launch angle, initial height, and gravity, you can instantly observe how these variables affect a projectile’s trajectory, range, and flight time.

The simulator provides a dynamic, real-time graph of the projectile’s path, along with precise calculations for key metrics such as maximum height and horizontal range. It’s a perfect tool for educational purposes, allowing for quick experimentation and a deeper grasp of how the equations of motion translate into real-world behavior.

How to Use the Tool

1. Adjust the Inputs: Use the sliders or the number input boxes on the left to set the desired values for initial velocity ($v_0$), launch angle ($\theta$), initial height ($h_0$), and gravity ($g$).
2. Observe the Trajectory: As you change the input values, the graph on the right will update in real-time, showing the new path of the projectile. The red dot represents the projectile’s current position.
3. View the Results: The calculated values for horizontal range, maximum height, and time of flight are displayed below the graph. These values are updated continuously.
4. Experiment: Try different combinations to see their effect. For example, a 45° launch angle gives the maximum range on a flat surface ($h_0 = 0$). Increasing the initial height will increase the range, while increasing the initial velocity will increase both the range and maximum height.
5. Reset: Click the “Reset” button to clear all inputs and start over with the default values.

Underlying Physics Formulas

The simulator uses the following standard kinematic equations for constant acceleration, where $t$ is time:

• Horizontal Position: $x(t) = v_0 \cos(\theta)t$
• Vertical Position: $y(t) = h_0 + v_0 \sin(\theta)t – \frac{1}{2}gt^2$
• Horizontal Velocity: $v_x(t) = v_0 \cos(\theta)$ (Constant)
• Vertical Velocity: $v_y(t) = v_0 \sin(\theta) – gt$
• Time of Flight ($t_{flight}$): $t_{flight} = \frac{v_0 \sin(\theta) + \sqrt{(v_0 \sin(\theta))^2 + 2gh_0}}{g}$
• Maximum Height ($H_{max}$): $H_{max} = h_0 + \frac{(v_0 \sin(\theta))^2}{2g}$
• Horizontal Range ($R$): $R = v_0 \cos(\theta) \cdot t_{flight}$

These equations are the basis for all the calculations and the trajectory you see on the graph. We hope this tool helps you explore the fascinating world of physics!