Vertical Throw Calculator

Pick throw up or throw down and enter v₀. On the way up the calculator computes H_max = v₀²/(2g). 15 m/s is about 11.5 m and 1.53 s to the peak, vertical only, no range Z.

Inputs

Result

Throw direction, Initial speed v₀, Time t (optional) and g (default 9.81). The result shows up here.

How it works

H_max h v₀ h = v₀t − ½gt²
Upward throw: climb to H_max, then fall back to the launch level.

A vertical throw moves only on the vertical. Upward: v(t) = v₀ − g t, h(t) = v₀ t − ½ g t². At the peak v = 0, so H_max = v₀²/(2g) and climb time is v₀/g. 15 m/s is H_max ≈ 11.5 m and about 1.53 s to the peak. Return to launch level takes twice that, about 3.06 s. At 10 m/s after 1 s: v ≈ 0.19 m/s, h ≈ 5.1 m.

Fields: direction, v₀, optional t, empty g = 9.81. Downward, H_max is not a climb peak: you already start with speed down. Blank t means you want H_max and the characteristic times, not the state at time t. A comma and a period are the same v₀: 12,5 and 12.5.

v₀ must be a number. Zero m/s up is no throw. A time longer than the return shows the formula state; ground impact at h = 0 uses flight time 2 v₀/g when launch was at ground level.

A drop from v₀ = 0 lives on the free-fall page: there you type t and get h. A nonzero angle is on full projectile. This calculator has no Z, because there is no horizontal component.

v₀ is squared in H_max. Twice the speed is four times the peak, not two. 30 m/s up is about 45.9 m, not 23 m.

Pick up, type 15, leave t empty, click Calculate, and match H_max ≈ 11.5 m. Then add t = 1 at v₀ = 10 and compare v and h. The header symbol does not toss water.

Drop from rest: free fall. Arc with an angle: full projectile.

How to use

  1. Pick direction: up or down. Up gives you H_max and climb times. Down computes a flight with v₀ already pointing down.
  2. Type v₀ in m/s, for example 15. That is speed at launch, not a mean over the climb.
  3. Leave t blank unless you want v(t) and h(t) at a chosen instant. Leave g blank for Earth.
  4. Click Calculate. 15 m/s up gives H_max ≈ 11.5 m and about 1.53 s to the peak. At 10 m/s after 1 s: v ≈ 0.19 m/s, h ≈ 5.1 m.
  5. A drop from rest lives on free fall. An arc with an angle is on full projectile. Here it stays vertical only.

Formula

Upward throw (axis upward):

v(t) = v0 − g t

h(t) = v0t − ½g t2

Hmax = v02 / (2g), time to apex tup = v0/g, round-trip flight time 2tup.

H_max and v₀ in a vertical throw

A vertical throw has no range Z. 15 m/s upward is H_max ≈ 11.5 m and about 1.53 s to the apex; return to launch level takes twice that.

H_max
Peak on an upward flight: H_max = v₀²/(2g). 30 m/s is about 45.9 m, four times the 15 m/s peak, not two.
v₀
Initial speed. Zero m/s upward is no throw. At 10 m/s after 1 s: v ≈ 0.19 m/s, h ≈ 5.1 m.
t
Optional time. Blank t means you want H_max and the landmark times, not the state at instant t.
g
Blank = 9.81. Formulas: v(t) = v₀ − g t, h(t) = v₀ t − ½ g t², axis upward.
v(t)
Speed at time t on an upward flight. At the apex v = 0. Downward, H_max as a climb peak does not apply.

Real-life examples

Example 1

v₀ = 15 m/s, upward, g = 9.81 m/s².

Example 2

v₀ = 10 m/s, t = 1 s: where is it after one second?

Example 3

v₀ = 8 m/s almost straight up.

Example 4

Water leaves the nozzle upward at 12 m/s.

Example 5

v₀ = 5 m/s downward, t = 1.5 s.

Example 6

v₀ = 15 m/s upward, g = 1.62 m/s².

Example 7

Takeoff speed 4 m/s upward: center-of-mass H_max.

Example 8

v₀ = 20 m/s upward: full flight time.

Ways to use this calculator

  • You find how high water goes at 15 m/s straight up.
  • You check v and h after 1 s at v₀ = 10 m/s.

Related calculators

Energy at the top
Potential energy at the highest point of the flight.
Energy at launch
Kinetic energy at launch, from mass and v₀.
Drop with zero launch speed
A horizontal throw: no upward launch speed.
Horizontal Throw
Range of a horizontal throw from height h
Projectile Max Height
The highest point of a projectile arc.

All physics calculators

Frequently asked questions

What is H_max at 15 m/s up?

H_max = 225 / 19.62 ≈ 11.5 m. Time to peak ≈ 1.53 s. Return to launch level ≈ 3.06 s.

Do I have to type time t?

No. t is optional. Without t you get H_max and the characteristic times. With t you get v(t) and h(t).

Which units do I type?

v₀ in m/s, t in seconds, g in m/s². Result in meters. Empty g = 9.81.

What if v₀ = 0?

Up, that is no throw. For a drop from rest, open free fall: there t gives h = ½ g t².

How is down mode different from free fall?

Here you have a downward v₀. Free fall starts at v₀ = 0 and computes h from t. Different unknown, different card.

Where is horizontal range?

This calculator has no Z, because there is no horizontal component. Open horizontal throw or full projectile.

Is there air drag?

No. Classroom vacuum model. In air the peak sits lower than 11.5 m at 15 m/s.

What if t is longer than the return?

The result is the formula state. Ground impact at h = 0 uses flight time 2 v₀/g when launch was at ground level.

How much after 1 s at 10 m/s up?

v ≈ 0.19 m/s, h ≈ 5.1 m. Still before the peak: climb time at 10 m/s is about 1.02 s.

Does a comma in 12.5 work?

Yes. 12,5 and 12.5 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.