Projectile Motion Simulator

Enter the initial speed, launch angle, starting height, and gravitational acceleration to calculate and chart the trajectory, range, and maximum height of projectile motion (ignoring air resistance). Try gravity on the Moon, Mars, or Jupiter, not just Earth.

Gravitational acceleration by celestial body

Body Gravity
Earth 9.8 m/s²
Moon 1.62 m/s²
Mars 3.71 m/s²
Jupiter 24.79 m/s²

Tips

  • This tool models idealized motion with no air resistance. Real balls and projectiles experience drag, so their actual range is shorter than the calculated value.
  • At a 45° launch angle, range is maximized for a given initial speed and gravity (assuming the launch height is 0).
  • Switch the gravity preset to the Moon or Mars to see how dramatically range and time of flight change for the exact same speed and angle.
  • A launch angle of 0° (horizontal launch) is useful for modeling something like throwing a ball horizontally from a height.

FAQ

When launching from ground level, the range is given by (initial speed² × sin(2 × angle)) ÷ gravity. Since sin(2 × angle) reaches its maximum value of 1 exactly when the angle is 45° (2 × 45 = 90°), that angle produces the greatest range of all.

With air resistance, an object follows an asymmetric path rather than a true parabola, and its range ends up shorter than this tool's calculated value. Drag matters more at higher speeds and for objects with a larger cross-section relative to their mass.

Earth's value uses the standard surface gravity (about 9.8 m/s²). The Moon, Mars, and Jupiter values are the published figures derived from each body's mass and radius.

Launching straight up (90°) means the horizontal velocity component is zero, so the range is zero — the object goes straight up and comes straight back down to the same spot.
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Side Note — Galileo's discovery of the parabola

It was Galileo Galilei, in the 17th century, who first showed mathematically that an object thrown at an angle traces a parabola — the same curve as a quadratic function's graph. Before that, thinking shaped by Aristotelian physics pictured a thrown object as traveling in a straight line before abruptly falling, with no notion of treating horizontal and vertical motion as independent.

Galileo's key insight was to combine two independent motions: constant-velocity horizontal motion (since no horizontal force acts on the object) and constantly-accelerating vertical motion (due to gravity). By considering these two motions together, he derived geometrically that the resulting path must be a parabola — a forerunner of the "resolving forces and motion into components" approach that would later become central to Newtonian mechanics.

Today, this same idea behind projectile motion underlies everything from artillery ballistics to rocket launch trajectories to the physics of a ball in sports. Rocket trajectory calculations, in particular, go far beyond this simple model — accounting for air resistance, Earth's rotation (the Coriolis effect), and the changing mass of the rocket as it burns fuel.