Example 1
v₀ = 20 m/s, α = 60°.
Projectile Max Height calculator
Enter v₀ and angle. The calculator computes peak height H_max = v₀² sin²(α)/(2g). 20 m/s at 60° is about 15.3 m, higher than the same 20 m/s at 45°, where the peak is about 10.2 m.
Result
Initial speed v₀, Angle α (degrees) and g (default 9.81). The result shows up here.
How it works
H_max depends on the vertical component: H_max = v₀² sin²(α) / (2g). At 20 m/s and 60°, sin 60° = √3/2, so H_max ≈ 15.3 m. At 45° the same v₀ gives about 10.2 m. At 30° you get about 5.1 m, a low, flat arc. At 90° the formula becomes a vertical throw: v₀²/(2g), about 20.4 m at 20 m/s.
Fields: v₀, α in degrees, empty g = 9.81. Result in meters. The formula is for a launch from ground level, no air drag. 60 in the field is 60°, not radians. A comma and a period are the same v₀: 15,5 and 15.5. At 15 m/s and 80° the peak is about 11.1 m.
α = 0° gives H_max = 0, because there is no upward component. That is not a horizontal throw from a tower: there h₀ is given, here launch is from the ground. Zero m/s gives H_max = 0. v₀ must be a number before 15.3 m can appear.
Compute Z next door. At 20 m/s and 45° the range is about 40.8 m and the peak is 10.2 m: different numbers, different cards. A nonzero h₀ is on the full-projectile page. There H_max = h₀ + v_y0²/(2g).
The closer you get to 90°, the higher the peak at the same v₀, but Z shrinks. 45° is a compromise for range, not for height. v₀ is squared: twice the speed is four times H_max.
Type 20 and 60, leave g empty, click Calculate, and match about 15.3 m. Then 20 and 45: about 10.2 m. The header symbol does not toss the ball under a ceiling.
Range on flat ground: Z. Launch from a tower: full projectile.
How to use
- In the first field enter v₀ in m/s, for example 20. That is speed at a ground-level launch.
- In the second field enter angle α in degrees, for example 60. The field wants degrees. 60 is 60°, not radians.
- Leave g blank for Earth. Click Calculate. 20 m/s and 60° give about 15.3 m. 20 m/s and 45° give about 10.2 m. 20 m/s and 30° give about 5.1 m.
- α = 0° gives H_max = 0. 90° becomes a vertical throw: at 20 m/s about 20.4 m. A tower launch lives on full projectile.
- Range Z is next door. At 20 m/s and 45° that is about 40.8 m, not 10.2 m. Here it stays the peak from ground level.
Formula
Hmax = v02 sin2(α) / (2g)
Helper: Z = v02 sin(2α) / g (level ground).
H_max, v₀, α, and g at the peak
This calculator computes peak height for a ground-level launch. 20 m/s at 60° gives H_max ≈ 15.3 m, higher than the same 20 m/s at 45° (about 10.2 m).
- H_max
- Peak height: H_max = v₀² sin²(α)/(2g). At 30° the same v₀ is about 5.1 m. At 90° the formula becomes v₀²/(2g), about 20.4 m at 20 m/s.
- v₀
- Launch speed from the ground. It is squared: twice v₀ is four times H_max. Zero m/s gives H_max = 0.
- α
- Angle in degrees. 60 in the field is 60°, not radians. α = 0° gives H_max = 0, because there is no upward component.
- g
- The constant in the 2g denominator. Blank = 9.81 m/s². A larger g cuts the peak at the same v₀ and α.
- sin²
- Sine squared of the angle in the H_max formula. At 60°, sin 60° = √3/2, hence about 15.3 m at 20 m/s.
Real-life examples
Example 2
v₀ = 20 m/s, α = 45°: H_max paired with Z.
Example 3
v₀ = 20 m/s, α = 30°: lower peak.
Example 4
v₀ = 15 m/s, α = 80°.
Example 5
v₀ = 12 m/s, α = 70°.
Example 6
v₀ = 8 m/s, α = 55°.
Example 7
v₀ = 20 m/s, α = 60°, g = 1.62 m/s².
Example 8
v₀ = 40 m/s, α = 75°.
Ways to use this calculator
- You check whether a 20 m/s, 60° arc fits under a 16 m ceiling.
- You compare H_max at 30°, 45°, and 60°.
Frequently asked questions
What is H_max at 20 m/s and 60°?
About 15.3 m. At 45° the same v₀ gives about 10.2 m. At 30° about 5.1 m. At 15 m/s and 80° about 11.1 m.
How is this different from range?
Here the main result is H_max, the peak of the arc. Z is on the range page. At 20 m/s and 45°, Z ≈ 40.8 m and H_max ≈ 10.2 m.
What about α = 90°?
The formula becomes a vertical throw: H_max = v₀²/(2g). 20 m/s is about 20.4 m. Range then drops to zero.
What about α = 0°?
H_max = 0. No vertical component. That is not a horizontal throw from a tower: you need h₀ on the full-projectile page.
Which units do I type?
v₀ in m/s, α in degrees, g in m/s². Result in meters. Empty g = 9.81.
What if v₀ = 0?
H_max = 0. Nothing to climb with. v₀ must be a number before 15.3 m can appear.
Is the angle in radians?
No. The field wants degrees. 60 is 60°, not 60 radians. There are no radians here.
Launch from a tower?
Open full projectile, that page has h₀. Here launch is from ground level. Peak from a tower is h₀ plus the climb from the vertical component.
Is there air drag?
No. Classroom vacuum model. In air the peak sits lower than these 15.3 m.
Does a comma in 15.5 work?
Yes. 15,5 and 15.5 are the same v₀. The calculator does not require a period.
Knowledge sources
The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.
Page updated in 2026.