A projectile motion simulator you can aim by hand: drag on the canvas to set the launch angle and speed, then let go and watch the parabola play out. Read off the range, the height of the apex and the total flight time as you tune gravity and angle — then switch on air resistance and watch the clean arc bend into a shorter, lopsided path. It all runs on your device.
Range R = v²·sin(2θ)/g versus launch angle, with no air resistance. The curve peaks at 45°; the dot marks your current angle. Complementary angles (like 30° and 60°) share a range.
In a vacuum, 45° flies the farthest — and complementary angles tie. Launch a shot at 45° and, with no air resistance, it covers more ground than any other angle. Even more surprising: any two angles that add up to 90° land in exactly the same spot. A flat, fast 30° shot and a high, lazy 60° shot travel the identical distance — one just spends much longer in the air. Turn on air resistance and both effects shift: the best angle drops below 45°, and the tidy symmetry breaks.
Reading the simulation
What every arrow, marker and curve on the screen is telling you.
1
The velocity splits into vx and vy
At the projectile the white arrow is the full velocity, and the two dashed arrows are its parts: a horizontal component vx = v·cosθ and a vertical component vy = v·sinθ. Watch them as the shot flies — with no air resistance the horizontal vx never changes, because nothing pushes sideways, while vy shrinks on the way up, hits zero at the top, then grows again pointing down. The two launch values are also written in the readouts so you can check vx and vy against the arrows.
2
The apex and landing markers on the parabola
The solid arc is the actual path; the faint dashed arc is the predicted flight. The blue marker sits at the apex — the highest point, where vy = 0 and height is (v·sinθ)²/2g — and the orange marker at the landing point gives the range. Because vx is steady while gravity works on vy, the curve is a symmetric parabola, and the Range, Max height and Flight time readouts update live to match the markers.
3
Air resistance shortens and steepens the path
Switch on Air resistance and the tidy parabola bends. Drag always points opposite the motion and grows with speed, so it bleeds off the fast horizontal drift: the shot falls short, the apex shifts left, and the descent becomes steeper and shorter than the climb. The path is no longer symmetric, so there is no neat formula — the simulator integrates the flight step by step and the markers move to where the projectile really peaks and lands.
How It Works
Two independent motions, one parabola — and a numerical solver for when the air pushes back.
1
Horizontal and vertical, kept separate
The launch velocity splits into a horizontal part, v₀·cosθ, and a vertical part, v₀·sinθ. Horizontally the projectile drifts at a constant speed, so x = v₀·cosθ·t. Vertically gravity pulls it down the whole time, so y = v₀·sinθ·t − ½g·t². The two axes never talk to each other — that independence is the heart of projectile motion.
2
Why the path is a parabola
Eliminate time between those two equations and y becomes a downward parabola in x. The apex sits where the vertical velocity hits zero, at height (v₀·sinθ)²/(2g); the projectile lands after a flight time of 2·v₀·sinθ/g, giving a range of v₀²·sin(2θ)/g. Because sin(2θ) peaks at θ = 45°, that angle throws the farthest — and sin(2θ) is unchanged when you swap θ for 90°−θ, which is why complementary angles match.
3
Adding air resistance
With drag switched on there is no neat formula. The acceleration becomes gravity minus a force that grows with the square of the speed and points opposite the velocity, a = g − k·|v|·v. The simulator steps this equation forward in small time increments, so the drawn path is the actual integrated flight and the readouts come straight from where the projectile really rises and lands.
What is projectile motion?
Projectile motion is the motion of an object launched into the air and moving under gravity alone. Its horizontal velocity stays constant while gravity pulls it downward, so the path traces a parabola. The launch speed and angle set how far and how high it goes, and the combination of steady horizontal drift and accelerating vertical fall produces the familiar curved arc.
What launch angle gives the maximum range?
In a vacuum, ignoring air resistance and launching from the same height it lands, a projectile travels farthest at a launch angle of 45°. Angles that add up to 90° — such as 30° and 60° — give the same range, one on a flat, fast arc and the other high and steep. With air resistance the best angle drops somewhat below 45°.
Does mass affect projectile motion?
No — mass does not affect projectile motion in a vacuum. All masses follow exactly the same path for a given launch speed and angle, because gravity gives every object the same acceleration regardless of its mass. A cannonball and a pebble launched identically trace identical arcs. Mass only starts to matter once air resistance is involved, since drag affects light objects more than heavy ones.
How does air resistance change projectile motion?
Air resistance pushes back against the direction of travel, so it shortens the range, lowers the optimal launch angle below 45°, and makes the trajectory asymmetric: the projectile rises on a wider curve and falls on a steeper, shorter one instead of a clean symmetric parabola. The drag modelled here is quadratic — the acceleration becomes a = g − k·|v|·v — so the retarding force grows with the square of the speed and bleeds off the most speed just after launch. With drag switched on there is no neat formula, so the flight is integrated numerically in small time steps rather than read from the analytic parabola.
Teaching or blogging about physics? You can embed this on your own site free — one line of code, no sign-up.
Classroom activity
A ready-to-assign lab using the simulation above. Free to use — nothing to install or sign up for, and no student data leaves the browser.
High school, College10–20 minInteractive simulation
Learning objectives
Explain why horizontal and vertical motion are independent.
Predict how launch angle affects range and maximum height.
Identify the angle that maximizes range with no air resistance.
Variables to change
Launch angle
Launch speed
Gravity
Air resistance (on/off)
Procedure
Set gravity to Earth (9.8 m/s²) and turn air resistance off.
Fix the launch speed. Fire at 15°, 30°, 45°, 60°, and 75°, recording the range each time.
Fire at 30° and 60° with the same speed and compare the two ranges.
Now turn air resistance on and repeat the 45° launch.
Observations
Record range, maximum height, and flight time for each launch angle in a table.
Questions
Which launch angle gave the greatest range with air resistance off?
How did the ranges at 30° and 60° compare? Why?
Does the projectile’s mass change its trajectory (air resistance off)? Explain.
What happened to the range and the best angle when you turned air resistance on?
At the top of the arc, what is the vertical velocity? The horizontal velocity?
Explanation
A projectile moves under gravity alone once launched. Horizontal velocity is constant while vertical velocity changes at g, so the path is a parabola. With no air resistance, 45° maximizes range and complementary angles share ranges. Air resistance shortens the range and lowers the optimal angle.
Answer key (for instructors)
1. 45° gives the maximum range when launched and landing at the same height with no air resistance.
2. They are equal. Complementary angles (adding to 90°) give the same range because the range formula depends on sin(2θ), and sin(60°)=sin(120°).
3. No. In a vacuum all masses fall with the same acceleration g, so the path depends only on launch speed and angle, not mass.
4. Range decreased and the angle giving maximum range dropped below 45°, because drag removes energy and grows with speed.
5. Vertical velocity is zero at the peak; horizontal velocity stays constant throughout (air resistance off).
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