rocket_launch

Projectile Motion Simulator

Explore 2D trajectories under different gravity environments. Adjust angle, speed, and drag — then launch to animate the flight.

Range

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Max height

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Flight time

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Impact speed

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Launch angle 45°

Initial speed 50 m/s

Air resistance

Projectile motion describes the curved path of an object launched into the air and subject only to gravity (and optionally drag). The key insight is that horizontal and vertical motion are independent: gravity only affects the vertical component, leaving horizontal velocity unchanged (without air resistance).

Horizontal position

x(t) = v₀ · cos(θ) · t

Constant velocity — no horizontal force acts on the projectile (without drag).

Vertical position

y(t) = v₀ · sin(θ) · t − ½gt²

Gravity decelerates upward motion, then accelerates the fall.

Maximum range

R = v₀² · sin(2θ) / g

Maximum range occurs at θ = 45°. Complementary angles (e.g. 30° & 60°) give the same range.

Maximum height

H = (v₀ · sin θ)² / 2g

Reached when vertical velocity = 0, at exactly half the total flight time.

Flight time

T = 2 · v₀ · sin(θ) / g

Twice the time to reach the apex. Inversely proportional to gravity.

Velocity components

vₓ = v₀cos θ | vᵧ = v₀sin θ − gt

Speed at impact equals launch speed (no drag). Direction is mirrored about the apex.

Anatomy of a trajectory

Independence principle Horizontal and vertical motions are completely independent. Gravity only acts vertically.

45° maximum range Without drag, a 45° launch always yields the greatest horizontal distance.

Symmetric trajectory The ascent and descent are mirror images in time. Impact speed equals launch speed.

Air resistance breaks symmetry Drag reduces range and height, and makes the descent steeper than the ascent.

Follow these steps to explore projectile motion. Each control directly corresponds to a physical variable in the equations — adjusting them in real time shows you exactly how they affect the trajectory.

Set the launch angle

Drag the Launch angle slider (1°–89°). Watch the green arrow on the canvas rotate and the trajectory reshape instantly. Try 45° for maximum range, or steep angles like 75° for high, short flights.

Adjust initial speed

The Initial speed slider controls launch velocity (5–120 m/s). Higher speed scales the entire trajectory proportionally — doubling the speed quadruples the range and height, since both depend on v₀².

Choose a gravity environment

Select a planet or body from the environment dropdown. Lower gravity (e.g. Moon at 1.62 m/s²) dramatically extends range and hang time; higher gravity (e.g. Jupiter at 24.79 m/s²) crushes the arc to a flat, fast path.

Toggle air resistance

Enable the Air resistance switch to add drag. Notice how the trajectory becomes asymmetric — a steeper descent, reduced range, and lower apex. The impact speed also drops well below the launch speed.

Launch and observe

Press Launch to animate the flight. The ball travels in real time along the computed arc, leaving a glowing trail. All four stat cards — Range, Max height, Flight time, Impact speed — update live as you move any slider.

Experiments to try

Complementary angles: Set 30° and then 60° at the same speed. Observe that both land at exactly the same range — a neat consequence of sin(2θ) = sin(2·(90°−θ)).

Moon vs Jupiter: Keep angle and speed fixed, then switch between Moon and Jupiter. The ratio of ranges closely matches the inverse ratio of their surface gravities (≈ 15×).

Air resistance asymmetry: Enable drag, then compare the trajectory shape to the clean parabola without it. Notice the apex shifts left — the drag-shortened ascent is slower than the steeper powered descent.