Projectile Motion Simulator

Created by: Ir. MD Nursyazwi

Simulation 1

Initial Velocity (m/s)

Launch Angle (degrees)

Initial Height (m)

Drag Coefficient

Simulation 2

Initial Velocity (m/s)

Launch Angle (degrees)

Initial Height (m)

Drag Coefficient

Show Velocity Vector

Show Force Vectors

Results 1

Time: 0.00 s

Distance: 0.00 m

Height: 0.00 m

Max Height: 0.00 m

Max Distance: 0.00 m

Total Flight Time: ---

Final Velocity: ---

Results 2

Time: 0.00 s

Distance: 0.00 m

Height: 0.00 m

Max Height: 0.00 m

Max Distance: 0.00 m

Total Flight Time: ---

Final Velocity: ---

How to Use the Simulator

Set Your Parameters: Use the input fields under "Simulation 1" (red) and "Simulation 2" (green) to set the initial velocity, launch angle, initial height, and drag coefficient for each projectile. You can use these two simulations to compare different scenarios side-by-side.
Start a Simulation: Click the Launch Simulation 1 or Launch Simulation 2 button to begin a flight path. The projectile's motion will be drawn on the canvas, and the data will update in real-time.
Visualize Vectors: You can toggle the visual representations of velocity and forces using the switches:
- **Show Velocity Vector**: Displays an arrow that shows the projectile's current speed and direction. The arrow's length corresponds to its speed.
- **Show Force Vectors**: Displays arrows for the forces acting on the projectile—a **blue arrow for gravity** and a **dark gray arrow for air resistance (drag)**. The length of these arrows represents the strength of the force.
Analyze Results: The "Results" panels on the right will display key data about each simulation. Once a projectile hits the ground, the final distance, flight time, and velocity will be shown.
Reset for a New Run: Click the Reset All button to clear the canvas and all data, allowing you to start a new simulation from scratch.

The Physics of Projectile Motion

The simulator is based on fundamental physics principles, breaking down a projectile's motion into its independent horizontal and vertical components. Here's a breakdown of the key equations used in the calculations.

Ideal Projectile Motion (No Air Resistance)

When the drag coefficient is set to zero, the simulator follows the classic kinematic equations. In this ideal scenario, the only force acting on the object is gravity, which causes a constant downward acceleration of $g = 9.81$ $m/s^2$.

Let's define the initial conditions:

Initial velocity: $v_0$
Launch angle: $\theta$
Initial height: $y_0$

The initial velocity is split into its horizontal and vertical components:

Initial horizontal velocity: $v_{x0} = v_0 \cos(\theta)$
Initial vertical velocity: $v_{y0} = v_0 \sin(\theta)$

Position and Velocity Equations over Time ($t$)

Horizontal Position: Since there is no horizontal force in this ideal case, the horizontal velocity is constant. $$x(t) = v_{x0}t$$
Vertical Position: The vertical position is affected by the initial vertical velocity and the constant acceleration due to gravity. $$y(t) = y_0 + v_{y0}t - \frac{1}{2}gt^2$$
Vertical Velocity: The vertical velocity changes linearly with time due to gravity. $$v_y(t) = v_{y0} - gt$$

Projectile Motion with Air Resistance (Drag)

When the drag coefficient is greater than zero, the simulation becomes more realistic. An additional force, the drag force ($F_D$), acts on the projectile in the opposite direction of its motion. This force is dependent on the projectile's velocity.

The formula for drag force is: $$F_D = \frac{1}{2} \rho A C_D v^2$$

Where:

$\rho$ (rho) is the density of air ($1.225$ $kg/m^3$).
$A$ is the cross-sectional area of the projectile.
$C_D$ is the drag coefficient (the value you input).
$v$ is the magnitude of the projectile's velocity.

Since the drag force is not constant, the acceleration is no longer constant either. The simulator handles this by using a numerical integration method. It takes tiny time steps and recalculates the drag force and the resulting acceleration at each point, providing a much more accurate and realistic simulation of the projectile's path.