Throw a ball and it does not travel in a straight line, and it does not simply drop. It arcs. That smooth curve is a parabola, and it comes from two motions happening at once: the ball keeps moving sideways at a steady speed while gravity pulls it down faster and faster. Set a speed and an angle in the launcher above, press Launch, and watch the arc trace out while the cards read off how far, how high and how long.
What is projectile motion?
Projectile motion is the motion of an object that has been thrown or launched and then moves under gravity alone, with air resistance ignored. Once it leaves your hand, the only force acting is gravity, so the object is in free fall sideways and downward at the same time. A kicked football, a thrown basketball, a long jumper and water arcing from a hose are all projectiles, and every one of them follows a parabola.
The launch velocity points in the direction you fire. It is convenient to split it into two pieces: a horizontal part and a vertical part. These two parts then behave completely independently, which is the key idea that makes the whole problem simple.
Two independent motions: steady horizontal, accelerating vertical
The horizontal and vertical motions do not affect each other:
- Horizontal: there is no horizontal force (air resistance is ignored), so the horizontal velocity vx = v0 cos θ stays constant the entire flight. The ball covers equal sideways distance in equal times, which is why the trail’s dots are evenly spaced left to right.
- Vertical: gravity acts straight down, so the vertical velocity vy = v0 sin θ shrinks on the way up, reaches zero at the top, then grows on the way down. This is exactly free fall, the same acceleration g that a dropped object feels.
Because horizontal speed is uniform and vertical speed changes at a constant rate, the combined path is a parabola. Watch the launcher: the ball slides across at a steady rate while it slows, stops climbing, and drops.
The projectile motion equations and formulas
For a projectile launched from ground level at speed v0 and angle θ, split the launch velocity into its two components and apply the free-fall relationships to the vertical part:
vx = v0 cos θ | vy = v0 sin θ
The vertical velocity runs out at the top and the flight is symmetric, so the object spends as long coming down as going up. That gives the three headline results:
time of flight: T = 2 v0 sin θ / g | max height: H = v0² sin²θ / (2g) | range: R = v0² sin 2θ / g
The range formula comes straight from the other two: horizontal distance is steady speed times time, R = vx × T = (v0 cos θ)(2 v0 sin θ / g), and 2 sin θ cos θ = sin 2θ. The launcher computes all three live, so you can treat it as a projectile motion calculator: set v0 and the angle, and read T, H and R before you even press Launch.
| Quantity | Formula | Depends on |
|---|---|---|
| Horizontal velocity vx | v0 cos θ | speed and angle (constant in time) |
| Vertical velocity vy | v0 sin θ | speed and angle (changes with time) |
| Time of flight T | 2 v0 sin θ / g | vy and g |
| Maximum height H | v0² sin²θ / (2g) | vy and g |
| Range R | v0² sin 2θ / g | speed, angle and g |
Why 45 degrees gives the greatest range
Range is proportional to sin 2θ. The sine function is largest, equal to 1, when its input is 90 degrees, so range is greatest when 2θ = 90°, that is θ = 45°. At exactly 45 degrees the range simplifies to its ceiling:
R_max = v0² / g
Low angles keep the ball fast sideways but leave it almost no time in the air; high angles buy lots of airtime but throw much of the speed straight up, wasting horizontal reach. Forty-five degrees is the balance point. The range-vs-angle graph beside the launcher makes it unmistakable: one smooth hump with its peak locked on 45.
Complementary angles land in the same place
Two angles that add up to 90 degrees, such as 30 and 60, or 20 and 70, are complementary, and at the same launch speed they produce the same range. The reason is again sin 2θ: since sin 2θ = sin 2(90° - θ), swapping θ for its complement leaves the range untouched. Turn on Show complementary angle and launch: the two balls arc very differently but land on the exact same range tick. The higher-angle shot climbs higher and stays airborne longer, yet covers no more ground.
Horizontal projectile motion (launched sideways)
A special case is a projectile launched horizontally from a height, with no upward velocity at all (launch angle 0). Now the vertical motion is pure free fall from rest, so the time to reach the ground depends only on the drop height, not on how fast it was thrown:
fall time from height h: t = √(2h / g) | horizontal distance: x = vx × t
This is why a bullet fired level and a bullet simply dropped from the same height hit the ground at the same instant: their vertical motions are identical, and the horizontal speed only decides how far downrange the fired one lands. The full free-fall relationships (v = √(2gh) and the rest) are covered in acceleration due to gravity.
Using the launcher as a projectile motion calculator
You do not have to launch anything to get numbers. The Range, Max height and Time of flight cards update the moment you move a slider, so the widget is a working calculator:
- Set Launch speed v0 and Launch angle with the sliders.
- Read Range, Max height and Time of flight from the cards.
- Switch Gravity to the Moon, Mars or Jupiter to see how a different g changes the same launch.
- Press Launch to watch the parabola, or use the presets to jump to classic cases.
The Time of flight card always shows the true physical time from the equations; the on-screen flight is played over a fixed short duration so that both a lazy lob and a flat rocket stay watchable.
Worked example: a ball kicked at 20 m/s
A ball is kicked at v0 = 20 m/s at 45 degrees (g = 9.8). First split the velocity:
vx = 20 cos 45° = 14.14 m/s | vy = 20 sin 45° = 14.14 m/s
Then apply the formulas:
T = 2 vy / g = 2(14.14) / 9.8 = 2.89 s | H = vy² / (2g) = 14.14² / 19.6 = 10.2 m | R = v0² sin 90° / g = 400 / 9.8 = 40.82 m
The launcher shows exactly these values in the Range, Max height and Time of flight cards.
For a complementary check at the same speed, compare 30 and 60 degrees: at 30 degrees, R = v0² sin 60° / g = 400(0.866) / 9.8 = 35.35 m; at 60 degrees, R = v0² sin 120° / g = 400(0.866) / 9.8 = 35.35 m. Equal range, because sin(2 × 30°) = sin(2 × 60°). The maximum heights differ, though: 5.1 m at 30 degrees versus 15.31 m at 60 degrees. And the absolute ceiling is the 45 degree case, R_max = v0² / g = 400 / 9.8 = 40.82 m, the peak of the range-vs-angle graph.
Projectile motion examples in real life
- A basketball shot arcs toward the hoop on a parabola; players aim for a fairly high angle so the ball drops down through the rim.
- A long jumper leaves the board at roughly 20 degrees, not 45, because a human cannot keep launch speed up at steep angles, so the real optimum is lower.
- Water from a fountain or hose breaks into countless little projectiles, each tracing its own parabola.
- A golf ball or football would follow these clean equations exactly if not for air, which adds drag and spin and flattens the real path.
Common mistakes and misconceptions
- “Heavier projectiles fall faster or travel less far.” The range, height and time equations contain no mass at all. Ignoring air, a heavy ball and a light ball launched identically follow the same path, because gravity gives every mass the same acceleration g. This is the same result as in acceleration due to gravity.
- “The horizontal and vertical motions affect each other.” They do not. Horizontal velocity stays constant (even spacing across the trail) while vertical velocity slows to zero at the top and then speeds up. Splitting the motion into independent parts is the whole trick.
- “A bigger angle always means more range.” No. Range peaks at 45 degrees and falls off on both sides, so a steep 75 degree lob undershoots 45, and 30 and 60 degrees tie.
- “Time of flight depends on how fast it goes sideways.” It does not. Airtime is set by the vertical launch speed, T = 2 v0 sin θ / g, so a 60 degree shot hangs far longer than a 30 degree shot at the same speed, even though they land the same distance away.
Keep exploring
The vertical half of every launch is ordinary free fall, laid out in acceleration due to gravity. See how a steadily changing velocity becomes a curved position graph in motion graphs, and why gravity gives the same acceleration to every mass in Newton’s laws of motion. The arc itself is the classic parabola from algebra, and you can put forces into a number with the F = ma calculator.