Catapult Simulation Calculator

The Physics Behind the Launch
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Catapult and Projectile Motion Simulator Online

Find out how far a projectile travels when launched by a catapult, a trebuchet or at a known speed. The simulator calculates the range, maximum height, flight time, impact speed and optimal launch angle, taking into account air resistance, wind and even the gravity of the Moon, Mars or Jupiter. It is free, works on phones and computers, and shows the step-by-step working for physics students and teachers.

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What is projectile motion?

Projectile motion is the movement of an object launched at a given speed and angle that is acted on only by gravity. The velocity splits into a horizontal part, which stays constant, and a vertical part, which gravity slows down and then reverses. The result is a parabola-shaped path. It is the basis for understanding everything from a soccer ball to a medieval catapult stone.

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How does a catapult work?

A catapult stores energy and transfers it to the projectile. This simulator offers three launch methods:

• Speed: you enter the launch speed of the projectile directly.

• Arm (mangonel): a force pushes the tip of the arm as it swings. The work done, W = F · L · φ, becomes kinetic energy. The simulator includes the arm's own mass, which also moves and takes part of that energy.

• Counterweight (trebuchet): a heavy weight drops from a certain height, and its potential energy, E = M · g · h, launches the projectile. The efficiency sets how much of that energy actually reaches the projectile.

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Projectile motion formulas

Without air resistance, the motion follows exact formulas:

• Velocity components: vx = v · cos θ, vy = v · sin θ

• Flight time from a height h: t = (vy + √(vy² + 2gh)) / g

• Horizontal range: x = vx · t

• Maximum height: h max = h + vy² / (2g)

When launched from the ground, maximum range is reached at 45°. When the launch starts above the ground, the optimal angle is lower than 45°, and the simulator finds it automatically.

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How does air resistance change the result?

In real life, air slows the projectile with a force that grows with the square of its speed: F = ½ · ρ · Cd · A · v². That is why a real object lands closer than it would in a vacuum, and its path is no longer a symmetric parabola: it comes down more steeply than it goes up. The simulator solves the flight numerically with the fourth-order Runge-Kutta method and lets you choose:

• The projectile shape (sphere, cube, cylinder, cone or streamlined body), which sets its drag coefficient.

• The projectile diameter.

• The air density: 1.225 kg/m³ at sea level and around 1.0 kg/m³ in high-altitude cities such as Denver or Mexico City, where projectiles fly farther.

• Wind, as a tailwind or a headwind.

The chart compares the trajectory with and without air, and the result shows how much range is lost to air resistance.

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What about other planets?

With the same launch speed, a projectile travels about 6 times farther on the Moon than on Earth, because lunar gravity is 1.62 m/s² compared with 9.81 m/s². On Jupiter, at 24.79 m/s², it covers only about 40 % of the distance. It is an easy way to see how gravity controls motion.

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What is this simulator useful for?

• Students: check projectile motion homework and see the full working.

• Teachers: show in class how angle, height, air and gravity affect a launch.

• School projects: design a catapult or trebuchet and predict its range before building it.

• Curious minds: experiment with launches on other planets or in strong wind.

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How accurate are the results?

Without air, the results match the exact projectile motion formulas. With air, the model includes drag and wind, but not projectile spin, axle friction or arm flex. A real catapult may therefore throw somewhat shorter than calculated. Adjusting the counterweight efficiency helps match the model to your own catapult.

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As a reminder, gravity is not uniform across the planet due to various factors; therefore, it is recommended to calculate it based on your specific location, which you can do here:
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