# Kinetic and potential energy, work and power calculator

> Kinetic energy (½mv²), potential energy (mgh), spring energy, work (Fd cos θ) and power (W/t), solved for any variable, plus the speed after a drop.

Interactive version: https://www.calcopenly.com/science/work-energy-power-calculator
Subject: Science calculators

Six formulas cover the common mechanical-energy problems: kinetic energy ½mv², gravitational potential energy mgh, spring energy ½kx², work Fd cos θ, average power W ÷ t, and the speed after a frictionless drop, √(v₀² + 2gh). Pick one, choose the variable to solve for, and the others become inputs.

Energy comes out in joules, with the kilowatt-hour equivalent beside it from 1 Wh (3,600 J) up; 1 kWh = 3.6 MJ. The default, a 1,000 kg car at 20 m/s (72 km/h), carries 200,000 J, or 0.0556 kWh; at 40 m/s it would carry four times as much.

Potential energy is measured from whichever level you call h = 0, so only differences in height matter. The drop mode ignores friction and air resistance, which makes its speeds an upper limit for real falls.

## Inputs

- **Quantity** (options: Kinetic energy ½mv², Gravitational potential energy mgh, Spring energy ½kx², Work W = Fd·cos θ, Power P = W/t, Speed at the bottom of a drop)
- **Solve for** (options: Energy, Mass, Speed)
- **Solve for** (options: Energy, Mass, Height)
- **Solve for** (options: Energy, Spring constant, Extension)
- **Solve for** (options: Work, Force, Distance)
- **Solve for** (options: Power, Work or energy, Time)
- **Solve for** (options: Speed at the bottom, Drop height)
- **Energy or work**
- **Mass**
- **Speed**
- **Height**
- **Spring constant k**
- **Extension or compression x**
- **Force**
- **Distance moved**
- **Angle between force and motion**
- **Time**
- **Power**
- **Speed at the top**
- **Speed at the bottom**
- **Gravitational acceleration**

## Results

- Energy or work (J) — main result
- Power (W)
- Speed at the bottom (m/s)
- Mass (kg)
- Speed (m/s)
- Height (m)
- Spring constant (N/m)
- Extension (m)
- Force (N)
- Distance (m)
- Time (s)
- Energy or work (kWh)

## Formula

$$
KE = \tfrac12 mv^2,\quad PE = mgh,\quad E_s = \tfrac12 kx^2,\quad W = Fd\cos\theta,\quad P = \frac{W}{t},\quad v_b = \sqrt{v_0^2 + 2gh}
$$

## Worked examples

### 1000 kg car at 20 m/s

- Quantity: Kinetic energy ½mv²
- Solve for: Energy
- Mass: 1000 kg
- Speed: 20 m/s
- **Energy or work: 200,000 J**
- **Energy or work: 0.055556 kWh**
- Checked against: Python 3.8 decimal: ½ × 1000 × 20² = 200000 J = 0.0555… kWh

### Speed of a 0.145 kg ball carrying 100 J

- Quantity: Kinetic energy ½mv²
- Solve for: Speed
- Energy or work: 100 J
- Mass: 0.145 kg
- **Speed: 37.1391 m/s**
- Checked against: Python 3.8 decimal: v = √(2E/m) = √(200/0.145) = 37.1390676…

### 50 kg lifted 10 m

- Quantity: Gravitational potential energy mgh
- Solve for: Energy
- Mass: 50 kg
- Height: 10 m
- **Energy or work: 4,903.33 J**
- Checked against: Python 3.8 decimal: 50 × 9.80665 × 10

### Spring k = 200 N/m compressed 10 cm

- Quantity: Spring energy ½kx²
- Solve for: Energy
- Spring constant k: 200 N/m
- Extension or compression x: 10 cm
- **Energy or work: 1 J**
- Checked against: Python 3.8 decimal: ½ × 200 × 0.1² = 1 J

### 100 N at 60° over 5 m

- Quantity: Work W = Fd·cos θ
- Solve for: Work
- Force: 100 N
- Distance moved: 5 m
- Angle between force and motion: 60 °
- **Energy or work: 250 J**
- Checked against: Python 3.8 math: 100 × 5 × cos60° = 250 J

### Force perpendicular to motion does no work

- Quantity: Work W = Fd·cos θ
- Solve for: Work
- Force: 100 N
- Distance moved: 5 m
- Angle between force and motion: 90 °
- **Energy or work: 0 J**
- Checked against: W = Fd cos 90° = 0 (OpenStax UP1 §7.1)

## Questions

### What is the formula for kinetic energy?

Kinetic energy is KE = ½mv²; with mass in kilograms and speed in metres per second the result is in joules. Because speed is squared, doubling it quadruples the energy: a 1,000 kg car has 200 kJ at 20 m/s and 800 kJ at 40 m/s. Rearranged for speed, v = √(2KE ÷ m), so a 0.145 kg ball carrying 100 J moves at 37.1 m/s.

### What is the difference between work and power?

Work is energy transferred by a force, W = Fd cos θ, measured in joules; power is how fast that transfer happens, P = W ÷ t, measured in watts, where 1 W = 1 J/s. Lifting 50 kg through 10 m takes 4,903 J however it is done; doing it in 5 s needs 981 W, and spreading it over 60 s needs 81.7 W.

### How many joules are in a kilowatt-hour or a kilocalorie?

One kilowatt-hour is 3,600,000 J, because it is 1,000 W sustained for 3,600 s. One kilocalorie, the food Calorie, is 4,184 J, using the thermochemical calorie of exactly 4.184 J listed in NIST SP 811. The 200,000 J carried by a 1,000 kg car at 20 m/s is therefore 0.0556 kWh, or about 47.8 kcal.

### How fast does an object hit the ground when dropped?

Without air resistance the impact speed is v = √(2gh), whatever the mass. A drop from 20 m ends at 19.81 m/s (71.3 km/h); a drop from 5 m ends at 9.90 m/s. Speed grows with the square root of height, so four times the height only doubles the speed. Air drag makes real speeds lower, most of all for light objects and long falls.

### How much energy does a spring store?

A spring stores E = ½kx², where k is the spring constant in N/m and x the extension or compression in metres. A 200 N/m spring compressed by 10 cm stores 1 J; compressed by 20 cm it stores 4 J. The formula holds only within the elastic range, where Hooke's law F = kx applies and the spring returns to its original length.

### How accurate is the kinetic and potential energy, work and power calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 8 worked examples whose answers come from independent sources; for example, “1000 kg car at 20 m/s” is checked against Python 3.8 decimal: ½ × 1000 × 20² = 200000 J = 0.0555… kWh.

### Where does the method come from?

OpenStax University Physics Volume 1, ch. 7 Work and kinetic energy; ch. 8 Potential energy and conservation of energy; HyperPhysics — Work, energy and power.

## Sources

- [OpenStax University Physics Volume 1, ch. 7 Work and kinetic energy; ch. 8 Potential energy and conservation of energy](https://openstax.org/books/university-physics-volume-1/pages/7-introduction)
- [HyperPhysics — Work, energy and power](http://hyperphysics.phy-astr.gsu.edu/hbase/work.html)