Library / Mechanical / MEC-10

Mechanical

Projectile height at a given time

Height of a launched object during its upward flight, with no air resistance.

Ref: MEC-10
Rev: v1
Field: Mechanical
Result: y [m]

x = x_{0} + u_{x} t, y = y_{0} + u_{y} t - \frac{1}{2} g t^{2}

Up is positive, uniform gravity, no air resistance. The launch velocity resolves into u_x = v₀cosθ and u_y = v₀sinθ. The calculator gives y on the way up (t not beyond the apex time u_y/g).

Horizontal motion is steady; vertical motion is slowed by gravity, so the path is a parabola.

Calculator

Solve for

yHeightComputed

v₀Launch speed

m/s

5300

θLaunch angle

585

tTime since launch

0.110

y₀Launch height

0100

Horizontal distance x − x₀34.47 m
Vertical velocity4.574 m/s

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Height (m) current value 19.39 m

tTime since launchKey driver

−29620.4

v₀Launch speed

−4.71280

y₀Launch height

17.9118

θLaunch angle

−5.6135.3

The apex is reached when the vertical velocity is zero, at t = u_y/g.
Air resistance, ignored here, reduces both range and height for real projectiles.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m)	Local step	Change in result
t Time since launch	0.1 to 10 s	−296	20.4	316.4	+10%	+2.97 %
v₀ Launch speed	5 to 300 m/s	−4.712	279.7	284.4	+10%	+14.9 %
y₀ Launch height	0 to 100 m	17.89	117.9	100	+10%	+0.773 %
θ Launch angle	5 to 85 °	−5.61	35.3	40.91	+10%	+12 %

Sweep of Time since launch

Time since launch (s)	Height (m)
0.1	3.379
1.338	18.52
2.575	18.64
3.813	3.748
5.05	−26.16
6.287	−71.1
7.525	−131
8.762	−206
10	−296

Sweep one variable

Vary

Height (m)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Height

Result · m

What the equation gives you. Everything else feeds it.

v₀

Launch speed

m/s · typical 5 to 300

Speed at launch. Raises the height through the vertical component.

Launch angle

° · typical 5 to 85

Angle above the horizontal. A steeper angle gives a larger vertical component.

Time since launch

s · typical 0.1 to 10

Time since launch (up to the apex). Height rises with time until the apex.

y₀

Launch height

m · typical 0 to 100

Height of the launch point. Adds directly.

Worked example with the values above

Vertical launch component: u_y = 30 sin 40° = 19.28 m/s.
Rise: u_yt − ½gt² = 28.93 − 11.03 = 17.89 m.
y = y₀ + rise = 19.39 m.

Assumptions and limits

Uniform gravity.
No air resistance or spin.

Common mistakes

Using degrees in a calculator set to radians.
Taking g as positive when up is positive.

Statistics and uncertainty

Angle errors matter more at steep angles; a 1° error at 40° changes the height by about 3%.

Where it is used

Trajectory and clearance checks.
Launch tests and ballistic estimates.

Maths you need

Resolving vectors: sin and cos components
Constant acceleration: s = ut + ½at²

Related formulas

Practise this formula