Article — Maximum Height Projectile Motion Calculator
Maximum Height Projectile Motion Calculator
Maximum height in projectile motion is H = v₀² × sin²(θ) / (2g). A ball thrown at 20 m/s at 45° on Earth reaches 10.2 m — about a three-story building. The same throw at 60° goes 50% higher because sin² grows faster than the loss in horizontal velocity matters for vertical reach.
What maximum height means in projectile motion
A projectile is anything in motion under gravity alone after release — a thrown ball, a fired arrow, a kicked soccer ball. Air resistance is normally neglected in the ideal model. The projectile's vertical velocity starts as v₀ sin(θ) upward, slows under gravity, hits zero at the peak, and then becomes downward as it falls. The peak is the maximum height.
Horizontal velocity is constant the whole time (still no air resistance, no friction). So at the peak the projectile is still moving forward at v₀ cos(θ) but with zero vertical velocity. That is the instant of maximum height. Galileo first separated the motion into independent horizontal and vertical components around 1600 — the insight that makes projectile motion tractable.
Galileo's Two New Sciences (1638) proved that the projectile path is a parabola. Aristotle had taught for nearly two thousand years that projectiles travel in a triangular path — straight forward, then straight down — and Galileo's parabolas overturned that received wisdom in a single page of geometry.
The maximum height projectile formula
The peak equation is H = v₀² sin²(θ) / (2g). Three inputs: initial speed, launch angle, gravity. One output: height above launch point. If the projectile starts at h₀ above ground, the absolute maximum height is h₀ + H.
Two related quantities follow immediately. Time to peak: t_peak = v₀ sin(θ) / g. Total flight time (for h₀ = 0): t_flight = 2v₀ sin(θ) / g — exactly twice the peak time, because the trajectory is symmetric. Horizontal range: R = v₀² sin(2θ) / g, which peaks at θ = 45°.
15° H = 1.37 m30° H = 5.10 m45° H = 10.20 m60° H = 15.30 m75° H = 19.04 m90° H = 20.39 mHow launch angle changes max height
Maximum height depends entirely on the vertical component of launch velocity. Horizontal velocity has no influence — it does not interact with gravity. So sin²(θ) drives the height curve: 0 at θ = 0, peaks at θ = 90° (straight up). At 45° you get sin² = 0.5, so half of the maximum-possible peak; at 60° you get sin² = 0.75, three-quarters.
Range tells a different story. R uses sin(2θ), which peaks at θ = 45°. The trade-off is built in: low angles give long range but low peak; high angles give high peak but short range. A 45° launch is the sweet spot for maximum range, not maximum height.
Maximum height for different speeds
Height scales with v₀². Double the speed, the max height quadruples. A baseball pitcher throws fastballs around 40 m/s; vertically (and ignoring drag) that would peak at 81 m. A soccer goal kick averages 25 m/s and peaks around 32 m. A golf drive leaves the clubface at roughly 75 m/s; without air resistance the ball would rise above 280 m, but drag and lift cut it to a real peak near 30–50 m.
Vertical launches are the simplest case: θ = 90°, sin = 1, so H = v₀² / 2g. A stone thrown straight up at 10 m/s reaches 5.1 m. The 30 m/s rocket fired straight up clears 45.9 m before falling back. Equation is symmetric: time up equals time down, and impact speed equals launch speed.
Projectile motion on other planets
Lower gravity, higher peaks. The relationship is inversely proportional: H scales as 1/g for fixed v₀ and θ. Moon gravity (1.62 m/s²) is 16.5% of Earth's, so a throw on the Moon goes 6.1× higher. Mars gravity (3.72) is 38% of Earth's, so a Mars throw goes 2.6× higher. Jupiter at 24.8 is 2.5× Earth's, so a throw there reaches only 40% of its Earth peak.
The famous Apollo 14 golf shot illustrates this. Alan Shepard's one-handed swing with a 6-iron, hampered by his pressurized suit, sent the ball "miles and miles" in his words — in reality, modern analysis of the footage suggests about 40 yards, still far more than the same swing would produce on Earth where air drag plus the suit constraint would have killed it.
To estimate maximum height fast, use H ≈ v₀y² / 20 in SI units (rounding g to 10). A 10 m/s vertical velocity gives H ≈ 5 m. The actual answer using g = 9.81 is 5.10 m — close enough for mental math.
Real-world projectile trajectories
Drag changes everything. A baseball thrown at 40 m/s at 45° "should" carry 163 m without air resistance. The actual carry distance is closer to 110 m because the drag force grows roughly with v² and pulls steeply on small high-speed objects. Lift from spin (Magnus effect) modifies the trajectory further; a backspun golf ball travels significantly farther than a no-spin shot.
For dense, slow-moving projectiles the no-drag model is accurate to within a few percent. Shot put (7.26 kg, low launch speed) is well-approximated by ideal projectile motion. Discus and javelin involve significant aerodynamic lift and are not. Bullets are nearly straight-line over short ranges but follow a noticeable arc at battlefield distances; military fire-tables embed precise drag corrections.
Common projectile motion mistakes
Three traps catch students and engineers alike.
If you forget to convert degrees to radians before applying sin(), the answer is gibberish. JavaScript's Math.sin takes radians; spreadsheets like Excel take radians; physics textbooks usually quote degrees. Always confirm.
Second, conflating maximum height and total flight time. Time to peak is half of total flight time only when h₀ = 0. From a 10 m tower the projectile spends less time going up than coming down. Third, ignoring h₀ in the range formula. The simple sin(2θ)/g formula only works for ground-to-ground launches; otherwise use the full time-of-flight expression with the quadratic discriminant.