Projectile Motion Calculator & Simulator

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🚀 Projectile Motion Calculator & Simulator

PROJECTILE MOTION

Projectile Motion Calculator & Real-time Ballistics Simulator
2D PHYSICS ENGINE ONLINE

Projectile Launch Parameters

(10 ~ 80 m/s)
m/s
(10 ~ 85 deg)
deg
(0 ~ 50 m)
m
(0.00 ~ 0.50)
Cd
(50 ~ 600 m)
m

Ballistic Scenario Presets

Real-time Trajectory & Velocity Vector Simulator

Aiming Complete
Real-time Flight Time
0.00 s
Real-time VelocityVx | Vy
0.00 m/s
Maximum Horizontal Range (R)
173.2 m
(Final Impact Velocity: 40.2 m/s)
Maximum Trajectory Height (H_max) 50.8 m
Total Flight Time (T_flight) 6.12 s
Target Miss Distance 23.2 m Deviation

Newtonian Equations of Motion

X(t) = V₀·cosθ · t [m]
Y(t) = h₀ + V₀·sinθ·t – 0.5·g·t² [m]
Disclaimer: The calculation results of this simulator are provided for educational and reference purposes only. For actual product design or manufacturing, please verify with the latest engineering standards and official design codes. The integrity of the calculated values is not guaranteed, and the developer and this blog assume no liability for any direct or indirect damages arising from their use.
💡 💡 Quick User Guide
  1. Input Initial Launch Conditions: Set the initial velocity (m/s), cannon launch angle (degrees), and initial height (m).
  2. Select Gravity and Air Resistance: Specify the gravitational acceleration of the celestial body—such as Earth (9.81 m/s²), Mars (3.71 m/s²), or the Moon (1.62 m/s²)—and the aerodynamic drag coefficient (Cd).
  3. Position the Target: Perform calculations taking into account the position of the red target placed on the screen.
  4. Real-Time Launch & Vector Monitoring: Click the [Launch!] button to observe the projectile flying along a smooth trajectory, reaching its peak, and view real-time vector resolution arrows for velocity.
📚 View Detailed Projectile Motion Formulas and Trajectory Equations ▼

1. Physical Fundamentals of Projectile Motion

An object launched in a 2D Cartesian coordinate system flies along a downward-curving parabolic path, influenced solely by gravitational acceleration. This motion is modeled by separating it into horizontal (X-axis) and vertical (Y-axis) components.

  • Horizontal (X) Motion: Since no external force acts on it (excluding air resistance), it maintains uniform linear motion. Vx = V0 × cos(θ)
  • Vertical (Y) Motion: Since gravitational acceleration (g) acts downward continuously, it undergoes uniformly accelerated vertical motion. Vy = V0 × sin(θ) - g × t

2. Formulas for Time of Flight (t_flight) and Range (R)

The equations of motion for a projectile launched from a height H_0 above the ground are derived as follows.

① Total Time of Flight (t_flight): Calculated using the quadratic formula for the point where the vertical height becomes Y(t) = 0.

t_flight = [V_0 × sinθ + √((V_0 × sinθ)^2 + 2 × g × H_0)] / g

② Maximum Horizontal Range (R): The horizontal distance traveled at a constant velocity during the total flight time.

R = V_0 × cos(θ) × t_flight  [m]

③ Peak Height of the Trajectory (H_max): The height reached at the exact moment when the vertical velocity Vy = 0.

H_max = H_0 + (V_0 × sinθ)^2 / (2 × g)  [m]

3. Physical Model Accounting for Air Drag

In real atmospheric conditions, drag force, which depends on air density and the shape of the projectile, acts to suppress the flight trajectory. In the simulation, the following external forces are calculated at each time step using a dynamical equation (Euler method) to reflect a realistic trajectory reduction.

Fx_drag = -1/2 × Cd × ρ × Vx × |V|    |    Fy_drag = -g - 1/2 × Cd × ρ × Vy × |V|

When such resistance is applied, the flight curve deviates from a perfectly symmetrical parabola, forming an asymmetrical trajectory that plunges steeply toward the ground just before impact.

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