Max Height Calculator
A ball kicked at an angle, a rocket leaving a launch pad, water arcing from a hose: all of them climb to a peak and then fall back down. This calculator finds that peak from the launch velocity and angle, with an optional starting height and a choice of gravity (Earth, Moon, Mars, Jupiter, or your own value), and it also returns the time to the apex, the total flight time, and the horizontal range.
Ideal projectile model: no air resistance. Results are the theoretical upper bound for real throws and launches, exact for the maths and optimistic for light or fast projectiles.
Quick Answer
A projectile launched at 25 m/s at a 60 degree angle from ground level on Earth peaks at 23.89 m, reaching the top after 2.21 s, staying airborne 4.41 s, and landing 55.17 m away. The governing formula is H = h0 + (v x sinθ)2 / (2g): only the vertical part of the launch velocity fights gravity, and it enters the formula squared, so doubling the launch speed quadruples the height.
- Quick Answer
- How to use this max height calculator
- How the maximum height formula works
- Worked example: a 25 m/s kick at 60 degrees
- How launch angle changes the flight
- The same launch on other worlds
- What other maximum height calculators miss
- Where the ideal model stops matching reality
- Frequently asked questions
- Add This Calculator to Your Website
How to use this max height calculator
Enter the launch velocity and angle, add a starting height if the launch begins above the landing level, and click Calculate. You get four results: maximum height, time to apex, total flight time, and horizontal range.
- Launch velocity is always in meters per second, and starting height is always in meters, even when you switch the results to feet. The units toggle converts the answers, not the inputs.
- Launch angle runs from 0 (flat) to 90 (straight up). Angles past 90 are rejected, because a launch angled back over your head is a different problem.
- Starting height can stay at 0 for a ground launch. Blank also counts as 0.
- Gravity defaults to Earth at 9.81 m/s2, the standard value listed by NIST for standard acceleration of gravity (9.80665 m/s2, rounded in the presets). Choose Custom value to model any other body or a classroom constant such as 10 m/s2.
How the maximum height formula works
The whole calculation rests on one fact: at the top of the arc the vertical speed is exactly zero. Gravity spends the climb erasing the vertical velocity, so the peak is wherever that budget runs out.
Vertical part: vy = v x sin(θ). Horizontal part: vx = v x cos(θ), which never changes.
Gravity removes g of vertical speed per second, so t = vy / g.
H = h0 + vy2 / (2g). The starting height h0 simply adds on top.
Fall time from the peak is √(2H / g). Total flight is rise plus fall, and range is vx x total flight time.
The derivation is the standard one taught from the kinematic equations: setting the vertical velocity vy - gt to zero gives the apex time, and substituting that time into y = vyt - (1/2)gt2 collapses to H = vy2 / (2g). OpenStax University Physics walks through the same derivation with the equations of motion for projectile motion, and Georgia State's HyperPhysics trajectory pages solve the same system interactively. Note the landing assumption baked into the flight time: the fall ends at height 0, so the starting height is measured above the landing level, not above sea level or the launch pad's base.
Worked example: a 25 m/s kick at 60 degrees
A soccer ball kicked at 25 m/s at a 60 degree angle from ground level on Earth:
- Vertical velocity: vy = 25 x sin(60 degrees) = 25 x 0.8660 = 21.65 m/s
- Maximum height: H = 21.652 / (2 x 9.81) = 468.75 / 19.62 = 23.89 m
- Time to apex: t = 21.65 / 9.81 = 2.21 s
- Total flight time: 2.21 + √(2 x 23.89 / 9.81) = 4.41 s (the fall from the peak takes the same time as the climb when launch and landing levels match)
- Horizontal range: 25 x cos(60 degrees) x 4.41 = 12.50 x 4.41 = 55.17 m
Run the same numbers in the calculator and you get the identical result. Two useful variants, also verified against the tool: switch the output to feet and the peak reads 78.38 ft with a range of 181.02 ft. Launch instead from a 10 m platform at 20 m/s and 45 degrees, and the peak rises to 20.19 m with a range of 49.08 m, because the starting height adds directly to the peak and stretches the fall. If your problem runs the other way, a dropped object with no launch at all, the height distance calculator covers drop height from fall time.
How launch angle changes the flight
Angle trades height against distance. For a fixed 20 m/s launch from ground level on Earth, every row below is a direct run of the calculator:
| Launch angle | Maximum height | Time to apex | Total flight time | Horizontal range |
|---|---|---|---|---|
| 30 degrees | 5.10 m | 1.02 s | 2.04 s | 35.31 m |
| 45 degrees | 10.19 m | 1.44 s | 2.88 s | 40.77 m |
| 60 degrees | 15.29 m | 1.77 s | 3.53 s | 35.31 m |
| 90 degrees | 20.39 m | 2.04 s | 4.08 s | 0.00 m |
Peak height at each angle (20 m/s, Earth):
Two patterns stand out. Angles equally spaced around 45 degrees (30 and 60) land at the same distance with very different arcs, because range depends on sin(2θ), which is symmetric around 45. And height depends on sin2(θ), so it keeps climbing all the way to 90 degrees, where the range collapses to zero. For longer horizontal problems where the arc is not the question, a distance calculator handles the straight-line side.
The same launch on other worlds
Gravity is the only environmental number in the formula, so swapping planets is just swapping g. The same 25 m/s launch at 60 degrees from ground level:
| Body | Gravity (m/s2) | Maximum height | Total flight time | Horizontal range |
|---|---|---|---|---|
| Earth | 9.81 | 23.89 m | 4.41 s | 55.17 m |
| Moon | 1.62 | 144.68 m | 26.73 s | 334.11 m |
| Mars | 3.71 | 63.17 m | 11.67 s | 145.89 m |
| Jupiter | 24.79 | 9.45 m | 1.75 s | 21.83 m |
Height, flight time, and range all scale with 1/g, which is why the Moon row is roughly six times the Earth row (9.81 / 1.62 is about 6.05). The preset gravities match the surface values published in NASA's planetary fact sheets for the Moon, Mars, and Jupiter. The acceleration side of the same physics, including how g itself is measured and used, is covered by the acceleration calculator.
Key takeaways
- Only the vertical component of velocity sets the height: H = h0 + (v sinθ)2 / (2g).
- Velocity is squared, so double the launch speed means four times the height.
- 45 degrees maximizes range for a ground launch; 90 degrees maximizes height and gives zero range.
- Starting height adds one for one to the peak and lengthens the fall, so it stretches the range too.
- Every result here ignores air resistance, so real peaks for light or fast projectiles land lower.
What other maximum height calculators miss
The top results for this keyword split into height-only tools and ground-only tools. The missing pieces are the same ones students and coaches actually ask about next.
- Omni Calculator's maximum height page answers the peak and only the peak: give it velocity, angle, and an optional initial height, and it returns hmax with no time to apex, no flight time, and no range, and no gravity choice beyond Earth's. Its worked example is in feet per second against a 13 ft fence, which is charming and also the whole scope.
- Calculator.info and AIOnlineCalculator projectile pages lead with the ground-level formula set (flight time = 2v sinθ / g, range = v2 sin(2θ) / g). Those formulas quietly assume the landing is at launch height, so a throw from a cliff or a balcony gets the wrong flight time and range unless you find the general case elsewhere on the page.
- Firgelli's projectile range calculator is thorough on range, including the elevated-launch formula, but it is organized around landing distance; the peak height is a side output, and gravity is a single typed constant with no planetary presets.
- The gap this page fills: starting height, selectable gravity with Moon, Mars, and Jupiter presets plus a custom value, and the full trajectory (peak, apex time, flight time, range) in one pass, with a meters or feet toggle on the results.
Where the ideal model stops matching reality
Air resistance is the big one. Drag grows with speed and with how light and large the projectile is, so a beach ball or a badminton shuttle peaks far below the number here, while a dense, slow toss tracks it closely. Real drag is also why falling objects stop accelerating at a ceiling speed, the subject of the terminal velocity calculator. Spin adds a second gap: a backspinning ball generates lift (the Magnus effect) and floats higher and longer than the no-spin model. Finally, the model treats gravity as constant and the landing as a flat plane at the starting height's base; over very long ranges or big height differences, both simplifications start to show.
Used inside its limits, though, the formula is exact, and it is the right first answer for homework, coaching estimates, fountain and sprinkler arcs, fireworks planning, and any launch where drag is a small correction rather than the main event. OpenStax College Physics uses the same ideal model for its projectile chapter for exactly that reason.
Frequently asked questions
H = h0 + (v x sinθ)2 / (2g), where v is the launch velocity, θ the launch angle, g gravity, and h0 the starting height (0 for a ground launch). Only the vertical component of the velocity, v x sinθ, works against gravity.
90 degrees, straight up. At 90 degrees, sin(θ) = 1, so the entire velocity works against gravity and H = h0 + v2 / (2g). The trade-off is zero horizontal range.
Yes. Drag eats into both the climb and the range, so a real ball peaks lower than the ideal formula predicts. The calculator shows the no-drag upper bound, which is close for short, slow, dense flights like a tossed ball and increasingly optimistic for light or fast projectiles.
It adds one for one: launching 10 m above the landing level raises the peak by exactly 10 m. It also lengthens the fall from the peak, which increases the total flight time and therefore the range.
Range needs both hang time (from the vertical component) and forward speed (from the horizontal component), and 45 degrees balances them for a ground launch. Height only cares about the vertical component, which is largest at 90 degrees.
About six times higher, because lunar gravity (1.62 m/s2) is about one sixth of Earth's (9.81 m/s2) and height scales with 1/g. The worked example launch of 25 m/s at 60 degrees peaks at 23.89 m on Earth and 144.68 m on the Moon. Pick Moon in the Gravity dropdown to run it.
Because velocity is squared in the formula: H depends on (v x sinθ)2. Double v and the squared term grows by a factor of four, so the peak quadruples. A 20 m/s straight-up launch reaches 20.39 m; at 40 m/s it would reach about 81.55 m.
Not in this model. Mass cancels out of the kinematic equations, so a heavy ball and a light ball launched at the same velocity and angle reach the same peak in a vacuum. In real air, mass matters indirectly: heavier, denser projectiles resist drag better and stay closer to the ideal result.
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References
- OpenStax, Rice University. University Physics Volume 1, Section 4.3: Projectile Motion. Accessed October 22, 2026.
- OpenStax, Rice University. College Physics 2e, Section 3.4: Projectile Motion. Accessed October 22, 2026.
- Nave CR, Georgia State University, HyperPhysics. Trajectories: projectile motion equations and solver. Accessed October 22, 2026.
- National Institute of Standards and Technology. Standard acceleration of gravity, CODATA value 9.80665 m/s2. Accessed October 22, 2026.
- NASA Science. Moon Facts: lunar surface gravity 1.62 m/s2. Accessed October 22, 2026.
- NASA Science. Mars Facts: Martian surface gravity 3.71 m/s2. Accessed October 22, 2026.
- NASA Science. Jupiter Facts: Jovian surface gravity 24.79 m/s2. Accessed October 22, 2026.