Potential Energy Calculator with Steps | PE = mgh, Any Units

A potential energy calculator finds the energy an object stores because of its position, using PE = m g h for height. A 2.50 kg book lifted 4.00 m onto a shelf stores 2.50 × 9.81 × 4.00 = 98.1 J.

Rule of thumb: lifting 1 kg by 1 m stores about 9.8 J. Double the mass or the height and the energy doubles. For a spring use PE = ½kx²; for charges use U = kq₁q₂ ÷ r.

A potential energy calculator works out the energy stored in a system because of the positions of its parts. Potential energy (PE) is stored energy, measured in joules (J). It is called "potential" because it can turn into motion later: a raised book can fall, a stretched spring can snap back, and two opposite charges can rush together.

The most common type is gravitational potential energy, the energy an object has because it sits at a height h above a chosen zero level. You calculate it as the product of mass, gravitational field strength and height: PE = m g h. The quantities used in the calculation are mass in kilograms, g in metres per second squared, and height in metres. The answer comes out in joules, because 1 kg × 1 m/s² × 1 m = 1 J.

A worked one-liner: lift a 2.50 kg book 4.00 m onto a high shelf with g = 9.81 m/s². PE = 2.50 kg × 9.81 m/s² × 4.00 m = 98.1 J. If the book falls off, that 98.1 J becomes kinetic energy, and the book hits the floor at about 8.86 m/s.

Who uses it: students in CBSE and ICSE Class 9 to 12, IGCSE, GCSE and A level, AP Physics 1 and 2, JEE and NEET, and first-year university physics and chemistry. Engineers use the same formula to size pumped-hydro storage, lifts and cranes. The idea of a stored, position-based energy grew through the 18th and 19th centuries; the Scottish engineer William Rankine named it "potential energy" in 1853.

Kinetic energy (KE = ½mv²) is the energy of motion. Potential energy is the energy of position. When friction is negligible, their sum, the mechanical energy, stays constant, which is how this page turns PE into a landing speed.

Pick a type of potential energy, choose what to solve for, then type your values in any units.

4.1 What the number means

Potential energy is the work you would have to do to move the object into its position, slowly and against the force. Lift a 2.50 kg book 4.00 m and you push up with about 24.5 N over 4.00 m, which is 24.5 N × 4.00 m = 98.1 J of work. That work is now stored. It is not lost, it is waiting.

Gravitational PE belongs to the object and the Earth together, not to the object alone. Physics teachers stress this because the energy is stored in the gravitational interaction between the two. For everyday problems you can say "the book has 98.1 J" as long as you remember the Earth is part of the system.

4.2 Is it a sensible size?

Use 10 J per kilogram per metre as a quick mental check. A 60 kg student climbing a 3 m flight of stairs gains about 1800 J. One food Calorie is 4184 J, so that climb uses less than half a Calorie of stored energy, although the body burns more because muscles are only about 20 to 25% efficient. A 1000 kg car on a 100 m hill stores about 1 MJ. A full tank of a pumped-hydro station stores millions of megajoules.

4.3 How it scales

PE = m g h is directly proportional to each of its three inputs. Double the mass, the energy doubles. Halve the height, the energy halves. Take the same book to the Moon, where g = 1.62 m/s², and the 4.00 m lift stores only 16.2 J. Elastic PE is different: it grows with the square of the stretch, so a spring stretched twice as far stores four times as much energy. Electric PE falls as 1 ÷ r, so halving the gap between two charges doubles their energy.

4.4 Sign and the zero level

You can put the zero of gravitational PE anywhere: the floor, a table top or sea level. A book on the floor has PE = 0 if the floor is your zero, and a negative PE if your zero is the table top. Only differences have physical meaning, so ΔPE = m g (h₂ − h₁) gives the same answer whatever zero you choose. For orbits and charges the zero is fixed at infinite distance, which is why gravitational PE far from the Earth, U = −G M m ÷ r, is always negative.

4.5 What the charts show

Chart 2 is a straight line through the origin for gravitational PE: a graph of PE against height has a gradient of m g. For a spring the same chart becomes a parabola, and the force-stretch chart is a straight line whose shaded area equals the stored energy. In every energy-during-the-fall chart, the green PE line and the blue KE line cross halfway down, and their sum (the dashed black line) stays flat. That flat line is conservation of energy.

4.6 When the simple model breaks down

PE = m g h assumes g is the same at every height. That is true to better than 0.3% for anything below about 20 km on Earth, so it is fine for buildings, mountains and aircraft. For satellites the error grows: at 408 km, the height of the International Space Station, m g h overestimates the lifting energy by 6.4%. Use the Orbital PE mode there. For springs the limit is the elastic limit; for falling objects it is air resistance, which always makes the real landing speed lower than √(2 g h).

4.7 Special cases

• h = 0: PE is zero by definition at the reference level.
• Negative h: the object is below your zero level, so PE is negative. That is allowed.
• Negative mass: not physical. The calculator stops and says so.
• Compression versus stretch: a spring squashed by 4 cm stores the same energy as one stretched by 4 cm.
• Opposite charges: negative U, a bound pair. Like charges give positive U and fly apart when released.

4.8 Where this leads next

Potential energy is the starting point for almost every energy problem. Add kinetic energy and you get the conservation of mechanical energy, which solves roller coasters, pendulums and projectiles. Add work done by friction and you get the full work-energy theorem. In chemistry, the same idea becomes Gibbs free energy, and in electricity it becomes voltage, which is simply electric potential energy per coulomb.

▶ Press Calculate above to fill all five write-ups with your own numbers: the full GRESS working, every unit, an uncertainty, a mark scheme and a plain-text version.

This flowchart shows how to calculate potential energy step by step, from a word problem to a checked answer with units. Every box uses the same worked example as the Excel, R and Python sections: a 2.50 kg book is lifted 4.00 m onto a shelf. Find its gravitational potential energy.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return 98.1 J for this book. Section 14 shows the same method written out by hand.

1. Choose the type of potential energy. Pick Gravitational PE for PE = mgh, Change in PE for a height difference, Elastic for springs, Electric for charges, Orbital for satellites, PE to KE for a falling object, or Cell potential for ΔG° = −nFE°.
2. Choose what to solve for. Click a Solve for chip. That field turns into a dashed answer box, and any value it does not need is greyed out.
3. Type the known values. Plain numbers, decimals and scientific notation all work: 2.50, 0.040 or 6.2e-3.
4. Pick the unit beside each value. Grams, pounds, feet, centimetres, microcoulombs, kilometres and more are converted to SI for you.
5. Set g for your syllabus. Use 9.81 for IGCSE, A level and most US courses, 9.8 for CBSE and AP, or 10 for quick estimates. Moon, Mars and Jupiter are also listed.
6. Pick the answer unit. The unit menu on the answer field sets the unit of the result, such as J, kJ, kWh, eV or ft·lbf.
7. Press Calculate. Nothing is worked out until you do, and changing any input clears the old answer so you never read stale numbers.
8. Read the answer box and the two checks. Units check ✓ confirms the units balance, and Verified confirms the answer substituted back into the formula.
9. Follow the working and the charts. The GRESS table, numbered steps, results table, diagram and four charts explain the answer.
10. Copy or download the working. Section 5 gives five ready-to-paste answers, and Download Doc or Download PDF saves the whole solution.

Excel has no potential energy function, and it does not need one. PE = m g h is a single multiplication, so one formula per row handles a whole class worth of problems. The two things worth doing properly are keeping g in its own cell, so you can switch between 9.81, 9.8 and 10 in one place, and putting the units in every column header.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

• Powers: =-3^2 returns 9 in Excel, not −9, because the minus sign binds first. Write =-(3^2) when a minus sign and a power meet, such as a negative height squared in a spring problem.
• Scientific notation: large answers such as 2.94E+13 J appear in E format. Use Format Cells → Scientific with two decimal places for three significant figures.
• Precision: Excel keeps about 15 significant digits, far more than any lab measurement, so round only in the final displayed cell with =ROUND(D2,1) or the number format.
• Solve for anything: for harder rearrangements use Data → What-If Analysis → Goal Seek. Set the PE cell to a target by changing the height cell.

9.4 Why Excel might disagree with R, Python or this page

So a "wrong" last digit is usually a different g, not a different method. The other common causes are rounding (Excel and R round 2.5 differently: Excel half away from zero, R half to even) and typing a constant to fewer figures than the CODATA value.

9.5 Charting it

1. Select the Height and PE columns.
2. Insert → Charts → Scatter with Straight Lines. Do not use the Line chart type, which spaces the x values evenly whatever they are.
3. For lab data, use Scatter only, then click a point → Add Trendline → Linear → tick Display Equation on chart.
4. The gradient of PE against height is m g. Divide it by the mass to get your measured g.
5. Label the axes with units: Height h (m) and Potential energy PE (J).

9.6 Errors you will hit

R works out potential energy in one line, PE <- m * g * h. The script below does more: it solves for mass and height, adds a spring and a hydrogen atom, prints every answer with its unit, runs a substitution check, and saves one 300 dpi figure of the energy changing during a fall. It uses base R only, so there is nothing to install.

10.1 The complete script

R, base only, saves potential_energy_plot.png

# ============================================================
# Potential energy calculator in R, with one 300 dpi figure
# StatsUnlock - https://statsunlock.com/potential-energy-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. Your values (SI units) -----------------------------
m <- 2.50      # mass in kg
h <- 4.00      # height in m
g <- 9.81      # gravitational field strength in m/s^2
               # use 9.80665 (standard), 9.8 (CBSE, AP) or 10 to match your book

# ---- 2. Gravitational potential energy ---------------------
PE <- m * g * h                  # joules (kg m^2 s^-2)
v  <- sqrt(2 * g * h)            # landing speed if dropped, m/s

# ---- 3. Rearrangements (solve for any variable) ------------
m_back <- PE / (g * h)           # mass from PE and h
h_back <- PE / (m * g)           # height from PE and m

# ---- 4. Elastic and electric PE, same idea -----------------
k <- 250; x <- 0.040             # spring: N/m and m
PE_spring <- 0.5 * k * x^2       # joules
eps0 <- 8.8541878188e-12         # F/m, CODATA 2022
kC <- 1 / (4 * pi * eps0)        # Coulomb constant, N m^2 C^-2
e  <- 1.602176634e-19            # elementary charge, C (exact)
U_H <- kC * (e) * (-e) / 52.9e-12   # hydrogen atom, joules

# ---- 5. Print the answers with units -----------------------
cat(sprintf("PE        = %.3g J\n", PE))
cat(sprintf("v (drop)  = %.3g m/s\n", v))
cat(sprintf("check m   = %.3g kg, check h = %.3g m\n", m_back, h_back))
cat(sprintf("spring PE = %.2f J\n", PE_spring))
cat(sprintf("hydrogen  = %.3g J = %.3g eV\n", U_H, U_H / e))
stopifnot(abs(m * g * h - PE) < 1e-12)          # substitution check
cat("Units: kg x m/s^2 x m = kg m^2/s^2 = J\n")

# ---- 6. One figure: energy during the fall -----------------
d  <- seq(0, h, length.out = 100)                # distance fallen, m
PEd <- m * g * (h - d); KEd <- m * g * d
png("potential_energy_plot.png", width = 7, height = 4.5, units = "in", res = 300)
plot(d, PEd, type = "l", lwd = 3, col = "#16a34a", ylim = c(0, PE * 1.08),
     xlab = "Distance fallen (m)", ylab = "Energy (J)",
     main = sprintf("2.50 kg book falling 4.00 m: PE = %.3g J", PE))
lines(d, KEd, lwd = 3, col = "#2563eb")
abline(h = PE, lty = 2, lwd = 2, col = "#0f172a")
legend("right", c("Potential energy", "Kinetic energy", "Total"),
       col = c("#16a34a", "#2563eb", "#0f172a"), lwd = 3, lty = c(1, 1, 2), bty = "n")
invisible(dev.off())

Output when run: PE = 98.1 J, v (drop) = 8.86 m/s, spring PE = 0.20 J, hydrogen = -4.36e-18 J = -27.2 eV. These match Sections 3, 7, 9 and 11.

10.2 What each part does

R traps: round(2.5) gives 2 because R rounds half to even; -2^2 is −4 in R (the opposite of Excel); log() is the natural log; and sin() takes radians. Keep SI units inside the calculation and convert only when you print.

10.3 What the figure shows

The green line is the book’s potential energy as it falls, from 98.1 J at the top to zero at the floor. The blue line is its kinetic energy, rising from zero to 98.1 J. They cross halfway down at 49.05 J each, and the dashed total stays flat at 98.1 J. That flat line is conservation of mechanical energy.

10.4 Common problems and fixes

10.5 Useful one-liners

# PE for a vector of heights
2.5 * 9.81 * c(1, 2, 4, 8)
# height needed to store 1 kJ with 5 kg
1000 / (5 * 9.81)
# change in PE from 1200 m to 1850 m for 72 kg
72 * 9.81 * (1850 - 1200)
# pounds and feet to SI, then PE
(150 * 0.45359237) * 9.81 * (35 * 0.3048)

Python does the same job with NumPy, SciPy and Matplotlib. SciPy’s constants module carries the CODATA values, so the electric and orbital examples use the same constants as this calculator. The script prints every answer with its unit and saves one 300 dpi figure.

Install once with pip install numpy scipy matplotlib. Google Colab already has all three.

11.1 The complete script

Python 3 with SciPy constants, saves potential_energy_plot.png

# ============================================================
# Potential energy calculator in Python, with one 300 dpi figure
# StatsUnlock - https://statsunlock.com/potential-energy-calculator/
# pip install numpy scipy matplotlib
# ============================================================
import numpy as np
import matplotlib.pyplot as plt
from scipy import constants as C

# ---- 1. Your values (SI units) -----------------------------
m = 2.50          # mass, kg
h = 4.00          # height, m
g = 9.81          # m/s^2; scipy's C.g is 9.80665 (standard gravity)

# ---- 2. Gravitational potential energy ---------------------
PE = m * g * h                    # joules
v = np.sqrt(2 * g * h)            # landing speed if dropped, m/s

# ---- 3. Rearrangements (solve for any variable) ------------
m_back = PE / (g * h)
h_back = PE / (m * g)

# ---- 4. Elastic, electric and orbital PE --------------------
k, x = 250, 0.040                                  # N/m, m
PE_spring = 0.5 * k * x**2                         # J
kC = 1 / (4 * np.pi * C.epsilon_0)                 # Coulomb constant
U_H = kC * C.e * (-C.e) / 52.9e-12                 # hydrogen atom, J
GM, R = 3.986004418e14, 6.371e6                    # Earth, m^3/s^2 and m
U_ISS = -GM * 1500 / (R + 408e3)                   # 1500 kg at 408 km, J
dG = -2 * C.physical_constants["Faraday constant"][0] * 1.10   # Daniell cell, J/mol

# ---- 5. Print the answers with units -----------------------
print(f"PE        = {PE:.3g} J")
print(f"v (drop)  = {v:.3g} m/s")
print(f"check m   = {m_back:.3g} kg, check h = {h_back:.3g} m")
print(f"spring PE = {PE_spring:.2f} J")
print(f"hydrogen  = {U_H:.3g} J = {U_H / C.e:.3g} eV")
print(f"ISS orbit = {U_ISS:.2g} J")
print(f"Daniell   = {dG / 1000:.3g} kJ/mol")
print(f"with C.g  = {m * C.g * h:.3g} J")
assert abs(m * g * h - PE) < 1e-12               # substitution check

# ---- 6. One figure: energy during the fall -----------------
d = np.linspace(0, h, 100)
fig, ax = plt.subplots(figsize=(7, 4.5))
ax.plot(d, m * g * (h - d), lw=3, color="#16a34a", label="Potential energy")
ax.plot(d, m * g * d, lw=3, color="#2563eb", label="Kinetic energy")
ax.axhline(PE, ls="--", lw=2, color="#0f172a", label="Total")
ax.set_xlabel("Distance fallen (m)")
ax.set_ylabel("Energy (J)")
ax.set_title(f"2.50 kg book falling 4.00 m: PE = {PE:.3g} J")
ax.legend(frameon=False)
fig.savefig("potential_energy_plot.png", dpi=300, bbox_inches="tight")

Output when run: PE = 98.1 J, v (drop) = 8.86 m/s, spring PE = 0.20 J, hydrogen = -4.36e-18 J = -27.2 eV, ISS orbit = -8.8e+10 J, Daniell = -212 kJ/mol, and with C.g = 98.1 J.

11.2 What each part does

Python traps: ^ is not a power; round(2.5) gives 2 (half to even); math.sin takes radians; and scipy.constants.g is 9.80665, which can differ in the third decimal place from a syllabus that uses 9.8.

11.3 What the figure shows

The same three lines as the R figure: potential energy falling, kinetic energy rising, and a flat dashed total at 98.1 J. If you add air resistance in your own version, the total line will slope down, and the gap is the energy lost as heat.

11.4 Common problems and fixes

11.5 Useful one-liners

# PE for many heights at once
import numpy as np; 2.5 * 9.81 * np.array([1, 2, 4, 8])
# spring constant from 0.20 J at 4.0 cm
2 * 0.20 / 0.040**2
# speed from 98.1 J and 2.50 kg
(2 * 98.1 / 2.50) ** 0.5
# joules to kWh
98.1 / 3.6e6

Run it in a terminal with python pe.py, in a Jupyter notebook cell, or in Google Colab without installing anything.

12.1 Constants used

Source: NIST, CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants (Mohr et al., 2025, reference 1).

12.2 Gravitational field strength

Sources: NASA Planetary Fact Sheets; IERS Conventions for Earth’s GM; IAU 2015 nominal solar radius. GM is known far more precisely than G or M separately.

12.3 Energy unit conversions

12.4 Typical potential energies

12.5 Typical spring constants

12.6 SI prefixes you will meet

This is the written method examiners expect, worked on the same example as Sections 7, 9, 10 and 11: a 2.50 kg book is lifted 4.00 m onto a shelf. Take g = 9.81 m/s². Find its gravitational potential energy, then the speed it would land at if it fell.

14.1 The written method (GRESS)

1. Given. m = 2.50 kg, h = 4.00 m, g = 9.81 m/s². Every value has a unit, and all are SI already.
2. Required. PE (joules), then v (m/s).
3. Equation. PE = m g h; then m g h = ½ m v², so v = √(2 g h).
4. Substitute, with units. PE = (2.50 kg)(9.81 m/s²)(4.00 m). v = √(2 × 9.81 m/s² × 4.00 m).
5. Solve. PE = 98.1 J. v = √78.48 = 8.86 m/s.
6. Check the units and the size. kg × m/s² × m = kg·m²/s² = J. About 10 J per kg per metre gives 2.5 × 4 × 10 = 100 J, so 98.1 J is the right size.
7. Write the answer with units and sensible significant figures. PE = 98.1 J and v = 8.86 m/s (3 significant figures, matching 2.50 and 4.00).

14.2 On a scientific calculator

Type the whole expression in one line instead of rounding in the middle. The brackets in "solve for height" matter: 98.1 ÷ 2.5 × 9.81 without them gives 384.9, which is wrong. Menu numbers for the CONST (physical constants) feature differ between models, so check the manual for yours.

14.3 Measure it yourself: a school lab practical

PE uses g = 9.81 m/s²: 52.0 × 9.81 × 3.60 = 1836 J, so 1840 J to 3 s.f. Paste the m and h columns into the Batch tab (for example 52.0, 3.60) to check every row at once.

14.4 Practice problems with answers

14.5 Mistakes that lose marks

• Leaving mass in grams or height in centimetres. Convert to kg and m before substituting, or the answer is 1000 or 100 times too big.
• Using the weight in newtons as the mass. If a question gives "a 50 N load", PE = W × h = 50 × h, with no extra g.
• Forgetting to square x in ½kx², or squaring the whole of ½kx.
• Measuring r from the surface instead of the centre in orbital problems.
• Dropping the minus sign on a loss of PE or on a bound (attracting) pair of charges.
• Using a different g from the question. If the paper says 9.8, use 9.8.
• Writing a final answer with no unit, which usually loses the mark even when the number is right.

Each potential energy formula has its own job. Picking the right one matters more than the arithmetic.

Use PE = m g h when:

• ✓ The object is near the surface of a planet, within about 10 km on Earth.
• ✓ You know, or can measure, the height above a fixed zero level.
• ✓ The force is gravity, not a spring or an electric force.
• ✓ You want the energy to lift something, or the energy it releases when it falls.

Use something else when:

• ✗ The height is large compared with the planet: use U = −G M m ÷ r (Orbital mode).
• ✗ The energy is stored in a stretched or squashed object: use ½ k x² (Elastic mode).
• ✗ The force is between charges: use k q₁ q₂ ÷ r (Electric mode).
• ✗ You want the energy of motion: use kinetic energy, ½ m v².
• ✗ Friction or air resistance does a lot of work: use the full work-energy theorem, not PE alone.

Real-world examples:

1. Pumped-hydro storage: water pumped uphill at night stores m g h; the head and the volume set how many gigawatt-hours the station holds.
2. Lifts and cranes: the motor must supply at least m g h, plus losses, to raise a load.
3. Roller coasters: the first hill stores the PE that carries the train round the whole track.
4. Archery and catapults: a drawn bow stores elastic PE that becomes the arrow’s kinetic energy.
5. Atoms and chemistry: electric PE between nucleus and electron sets ionisation energies; ΔG° = −nFE° sets what a battery can deliver.
6. Animal movement: kangaroos and wallabies store elastic energy in their tendons on each hop and get much of it back on the next one.

Assumptions

1. Uniform gravity (PE = m g h). g is taken as the same at every height. Check: height under about 10 km on Earth. If it fails, m g h overestimates the energy, by a fraction about equal to h ÷ R (6.4% at 408 km).
2. A fixed zero level. Heights are measured from one reference. Check: both heights in a change use the same zero. If they do not, ΔPE can come out with the wrong size or sign.
3. No friction or air resistance (PE to KE). All the PE becomes KE. Check: dense objects over short drops. If it fails, the formula overestimates the landing speed.
4. Hooke’s law (springs). Force grows in proportion to the stretch. Check: stay within the elastic limit. Beyond it, ½kx² overestimates the energy you can get back.
5. A massless spring. Check: spring mass small compared with the load. If not, some energy goes into moving the spring, so the load gets less than ½kx².
6. Point charges in a uniform medium. Check: sizes much smaller than r. Spread-out charges and changing media make the simple formula overestimate or underestimate, depending on the shape.
7. A spherical, non-rotating planet (orbital mode). Check: precision needed below about 0.5%. Earth’s rotation and flattening change effective surface g by up to about 0.5% between the equator and the poles.
8. Standard conditions (cell mode). E° and ΔG° apply at 1 mol/L and 1 bar. Real concentrations shift E through the Nernst equation, so the real ΔG can be larger or smaller.

Limitations

• It does not include relativity. For speeds above about 0.1c, use relativistic energy instead of ½mv².
• It does not model air resistance. A projectile or free-fall simulation with drag is the better tool for real falls.
• It treats each type of PE separately. Systems with gravity and a spring together, such as a bungee jump, need both terms added.
• It uses two-body formulas. Three or more charges or bodies need the PE of every pair added up.
• Cell mode reports standard values only. Non-standard cells need the Nernst equation.
• Since 2019 the vacuum permeability μ₀ is no longer exactly 4π × 10⁻⁷ N/A², and ε₀ is a measured value. The change only affects the tenth significant figure, so textbook answers using k = 8.99 × 10⁹ still agree to 3 figures.

A potential energy calculator answers one question: how much energy is stored in a system because of where its parts are? For a raised object that is m g h, so a 2.50 kg book on a 4.00 m shelf holds 98.1 J. The formula is short, but most lost marks come from the steps around it: units, the value of g, the zero level, and the sign.

Three habits fix nearly every error. First, convert to SI before you multiply, because grams, centimetres and feet are the source of most answers that are 100 or 1000 times off. Second, use the g your question gives, since 9.8, 9.81 and 10 change the last digit. Third, check the answer by substituting it back, which is exactly what the Verified badge does for you.

Potential energy also links to the rest of physics. Set the PE lost equal to the KE gained and you get the landing speed, v = √(2gh). Replace gravity with a spring and you get ½kx²; replace it with charges and you get kq₁q₂ ÷ r; go far from the Earth and you need −GMm ÷ r. In chemistry, the same idea becomes ΔG° = −nFE°.

Choose your type of energy, type your values in any units, and let this potential energy calculator show every step, the units check and a diagram drawn from your own numbers.

These peer-reviewed articles support the formulas, constants, teaching methods and common mistakes covered by this potential energy calculator. Every DOI was checked in October 2026.

1. Mohr, P. J., Newell, D. B., Taylor, B. N., & Tiesinga, E. (2025). CODATA recommended values of the fundamental physical constants: 2022. Reviews of Modern Physics, 97(2), 025002. https://doi.org/10.1103/RevModPhys.97.025002
2. Hecht, E. (2003). An historico-critical account of potential energy: Is PE really real? The Physics Teacher, 41(8), 486–493. https://doi.org/10.1119/1.1625210
3. Hecht, E. (2007). Energy and change. The Physics Teacher, 45(2), 88–92. https://doi.org/10.1119/1.2432084
4. Keeports, D. (2017). Locating gravitational potential energy. Physics Education, 52(1), 013007. https://doi.org/10.1088/1361-6552/52/1/013007
5. Mungan, C. E. (2005). A primer on work-energy relationships for introductory physics. The Physics Teacher, 43(1), 10–16. https://doi.org/10.1119/1.1845983
6. Jewett, J. W., Jr. (2008). Energy and the confused student I: Work. The Physics Teacher, 46(1), 38–43. https://doi.org/10.1119/1.2823999
7. Arons, A. B. (1999). Development of energy concepts in introductory physics courses. American Journal of Physics, 67(12), 1063–1067. https://doi.org/10.1119/1.19182
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