Example 1
h₀ = 0, α = 45° → Z = v₀²/g.
Z ≈ 40.8 m
20 m/s, 45°, h₀ = 0. Same range as Z = v₀²/g.
Projectile motion calculator
Enter v₀, angle, and launch height h₀. The calculator computes Z, H_max, and flight time, including a launch from a tower. At h₀ = 0, 20 m/s, and 45° you get the same range as the Z page, about 40.8 m.
Result
Launch speed v₀, Angle α (degrees), Launch height h₀ and g (default 9.81). The result shows up here.
How it works
Full projectile splits v₀ into vₓ = v₀ cos α and v_y0 = v₀ sin α. H_max = h₀ + v_y0²/(2g) when the vertical component points up. Flight time comes from h₀ + v_y0 t − ½ g t² = 0, then Z = vₓ t. At h₀ = 0 and 45°, Z ≈ v₀²/g: 20 m/s is about 40.8 m, same as the range page. At 25 m/s, 60°, and h₀ = 0, Z ≈ 55.2 m and the peak ≈ 23.9 m.
Fields: v₀, α in degrees, h₀ in meters (0 = landing level), empty g = 9.81. No air drag. 45 is 45°, not radians. h₀ cannot be negative. A comma and a period are the same h₀: 10,5 and 10.5.
α = 0° and h₀ = 8 m at 6 m/s is a horizontal throw from an edge: t ≈ 1.28 s, Z ≈ 7.7 m, the same model as the horizontal-throw page. At 15 m/s, 30°, and h₀ = 10 m, flight time is about 2.38 s, Z about 31 m, peak about 12.9 m.
The narrower Z and H_max cards stay at Δh = 0. Here it stays h₀. Blank h₀ is usually taken as 0, launch at landing level. v₀ must be a number. A 90° angle at h₀ = 0 returns at your feet, Z = 0.
Type 20, 45, and 0, leave g empty, click Calculate, and read Z, H_max, and time. Then 6, 0, and 8: the same flight as horizontal throw. The header symbol does not drop you off a roof.
The model is classroom vacuum: constant g, flat landing level. If you want only Z or only the peak with no tower, open the narrower card. Here three results at once, with h₀.
Z only on flat ground: range. Peak only: H_max.
How to use
- Type v₀ in m/s, for example 20. That is speed at launch, not a mean over the flight.
- Type angle α in degrees, for example 45. The field wants degrees. 0° and a nonzero h₀ is a throw from an edge.
- Type h₀ in meters. Zero means launch at landing level. 8 m is the edge in the example. Leave g blank for Earth.
- Click Calculate. Read Z, H_max, and flight time. 20 m/s, 45°, and h₀ = 0 give Z ≈ 40.8 m. 6 m/s, 0°, and 8 m give t ≈ 1.28 s and Z ≈ 7.7 m.
- Z only on the flat lives on the range page. Peak only, no h₀, lives on H_max. Here it stays the full flight, with or without a tower.
Formula
vx = v0 cosα, vy0 = v0 sinα
Hmax = h0 + vy02/(2g) (when vy0 > 0)
0 = h0 + vy0 t − ½g t2 → t, then Z = vx t
h₀, Z, and H_max on a tower launch
The full flight keeps launch height. At h₀ = 0, 20 m/s and 45°, Z ≈ 40.8 m, same as the range page; at h₀ = 8 m and α = 0° it matches a horizontal throw.
- h₀
- Launch height in meters. Blank is usually 0, landing level. Negative will not run. 8 m is the example edge.
- v₀
- Launch speed. The calculator splits it into vₓ = v₀ cos α and v_y0 = v₀ sin α.
- α
- Angle in degrees. 0° and a nonzero h₀ is a throw from an edge. 90° at h₀ = 0 returns at your feet, Z = 0.
- Z
- Range: Z = vₓ t, with t from h₀ + v_y0 t − ½ g t² = 0. At 25 m/s, 60°, and h₀ = 0, Z ≈ 55.2 m.
- H_max
- Peak: H_max = h₀ + v_y0²/(2g) when the vertical part goes up. At 20 m/s, 45°, h₀ = 0 that is about 10.2 m.
- vₓ
- Horizontal component, constant in vacuum. The result is vₓ as a number. Z = vₓ t after flight time is found.
Real-life examples
Example 2
v₀ = 15 m/s, α = 30°, h₀ = 10 m.
Z ≈ 31 m
15 m/s, 30°, h₀ = 10 m. Flight time about 2.38 s, peak about 12.9 m.
Example 3
α = 0, h₀ = 8 m, v₀ = 6 m/s.
Z ≈ 7.7 m
6 m/s, 0°, h₀ = 8 m. Horizontal edge throw, t ≈ 1.28 s.
Example 4
v₀ = 25 m/s, α = 60°, h₀ = 0.
Example 5
v₀ = 18 m/s, α = 30°, h₀ = 0.
Example 6
v₀ = 12 m/s, α = 20°, h₀ = 15 m.
Example 7
v₀ = 20 m/s, α = 45°, h₀ = 0, g = 1.62 m/s².
Example 8
v₀ = 8 m/s, α = 40°, h₀ = 2 m.
Example 9
v₀ = 16 m/s, α = 80°, h₀ = 0.
Ways to use this calculator
- You compute Z from a roof: v₀, α, and h₀ together.
- You check that h₀ = 0 matches the range page.
Frequently asked questions
How is this different from the range page?
This page has h₀. That page assumes Δh = 0 and the main result is Z alone. At h₀ = 0, 20 m/s, and 45° both cards give about 40.8 m.
What is Z at 20 m/s, 45°, h₀ = 0?
About 40.8 m, same as Z = v₀²/g. The peak is about 10.2 m. At 25 m/s, 60°, and h₀ = 0, Z ≈ 55.2 m.
Which units do I type?
v₀ in m/s, α in degrees, h₀ in meters, g in m/s². Empty g = 9.81. Results in meters and seconds.
Is blank h₀ zero?
Blank h₀ is usually taken as 0, launch at landing level. v₀ must be a number. Negative h₀ will not run.
α = 0° and h₀ = 8 m?
That is a horizontal throw from an edge. At 6 m/s: t ≈ 1.28 s, Z ≈ 7.7 m. You can also open the horizontal-throw page.
How much at 15 m/s, 30°, and h₀ = 10 m?
Flight time about 2.38 s, Z about 31 m, peak about 12.9 m. That pair sits on the example card.
Is the angle in radians?
No. The field wants degrees. 45 is 45°, not π/4. There are no radians here.
Is there air drag?
No. Classroom vacuum, constant g. In air, Z and H_max come out shorter and lower.
Where is H_max without h₀?
On the projectile-height page. Launch there is from ground level. Here H_max = h₀ + v_y0²/(2g) when the component points up.
Does a comma in 10.5 m work?
Yes. 10,5 and 10.5 are the same h₀. 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.