Formula of Height: h = ½gt², h = u²/2g and h = PE/mg

The formula of height in free fall is $h = \frac{1}{2}gt^2$ : an object dropped from rest falls a height equal to half of g times the square of the fall time. Time a stone dropped from a cliff, and this formula turns the seconds into metres.

Physics has a few other height formulas, each for a different kind of question. The greatest height of a ball thrown straight up is $h = \frac{u^2}{2g}$ . The height gained from potential energy is $h = \frac{PE}{mg}$ . And the height of a tree or building seen from a distance d at an angle of elevation θ is $h = d\tan\theta$ . This page works an example of each, using g = 9.8 m/s² and ignoring air resistance.

Quick facts

Free fall from rest
$h = \frac{1}{2}gt^2$
Vertical throw, top point
$h = \frac{u^2}{2g}$
From potential energy
$h = \frac{PE}{mg}$
From an angle
$h = d\tan\theta$
SI unit
metre (m)
Fall time
$t = \sqrt{\frac{2h}{g}}$

Height formula physics: four versions

Which height formula to use
Situation	Formula	Comes from
Dropped from rest, time known	$h = \frac{1}{2}gt^2$	Second equation of motion with u = 0 and a = g
Thrown straight up at speed u	$h = \frac{u^2}{2g}$	$v^2 = u^2 - 2gh$ with v = 0 at the top
Energy given	$h = \frac{PE}{mg}$	$PE = mgh$
Angle of elevation from distance d	$h = d\tan\theta$	Right angled triangle

The first two are kinematics, the third is energy and the last is geometry. They are linked: a ball dropped from the height reached in a vertical throw lands with the same speed u it was thrown with.

Formula of height examples

Example 1: a stone off a cliff

A stone dropped from the top of a cliff hits the sea 3.0 s later. How high is the cliff?

$h = \frac{1}{2}(9.8)(3.0)^2 = 4.9 \times 9.0 = 44.1\ \text{m}$

Example 2: a vertical throw

A juggler tosses a ball vertically at 14 m/s. How high does it rise above the hand?

$h = \frac{14^2}{2 \times 9.8} = \frac{196}{19.6} = 10\ \text{m}$

Example 3: from energy, starting in kilojoules

A climber of mass 50 kg gains 24.5 kJ of gravitational potential energy on a climb. Through what height did she rise?

Convert: 24.5 kJ = 24 500 J. $h = \frac{24\,500}{50 \times 9.8} = \frac{24\,500}{490} = 50\ \text{m}$ .

Example 4: the height of a tree

Standing 20 m from the foot of a tree, a student measures the angle of elevation of its top as 35°. Ignoring eye level, how tall is the tree?

$h = 20 \times \tan 35^\circ = 20 \times 0.700 \approx 14.0\ \text{m}$

Why height grows with the square of time

A falling stone gains 9.8 m/s of speed in each second it falls. The stone covers more ground in each second than in the one before, so the height grows faster than the time. Double the fall time and the height is four times larger. In the first second a dropped stone falls 4.9 m, by the end of 2 s it has fallen 19.6 m, and by 3 s it has fallen 44.1 m.

Turned around, the fall time grows only with the square root of the height. Dropping from four times as high only doubles the time. The formula of time page uses this form, t = √(2h/g).

Units and checks

Height is a length, so its SI unit is the metre and it has dimensions [M⁰ L¹ T⁰]. Check h = ½gt²: (m/s²) × s² = m. Check h = u²/2g: (m/s)² ÷ (m/s²) = m. If your rearranged formula does not reduce to metres, it is wrong. A quick sense check also helps: a three storey building is about 10 m tall, and a stone takes about 1.4 s to fall that far.

Height mistakes in exams

Forgetting to square t. h = ½gt² needs t², so 3 s gives 9 s².
Using h = ½gt² for a throw. It is only for a drop from rest. For a launch with speed u, use the full equation of motion.
Measuring from the wrong point. In a projectile problem, the maximum height is measured from the launch point.
Energy in kJ. Convert to joules before dividing by mg.

Projectile heights are on the maximum height of projectile formula page, energy heights on the potential energy formula page, and the full set of falling body equations under kinematics formulas.

Frequently asked questions

Is there one formula of height in physics?

No, it depends on the data. For a body dropped from rest, h = ½gt². For the top of a vertical throw, h = u² / 2g. From energy, h = PE / mg.

How do you find height from time?

If the object falls from rest, square the time, multiply by g and halve the result: h = ½gt². With g = 9.8 m/s², a 2.0 s fall is 19.6 m.

How do you find the height of a building with an angle?

Measure the distance d to its base and the angle of elevation θ to the top. Then h = d tan θ, plus the height of your eye if it matters.

What is the SI unit of height?

The metre (m), the SI unit of length.

Does mass affect the height an object falls in a given time?

No. In a vacuum a heavy ball and a light one dropped together stay side by side, and h = ½gt² contains no mass term.