# 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.

操作できるページ：https://www.calcopenly.com/ja/science/work-energy-power-calculator
分野：科学の計算機

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.

## 入力

- **Quantity** (選択肢：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** (選択肢：エネルギー, 質量, 速さ)
- **Solve for** (選択肢：エネルギー, 質量, 高さ)
- **Solve for** (選択肢：エネルギー, Spring constant, Extension)
- **Solve for** (選択肢：Work, 力, Distance)
- **Solve for** (選択肢：仕事率, Work or energy, 時間)
- **Solve for** (選択肢：Speed at the bottom, Drop height)
- **Energy or work**
- **質量**
- **速さ**
- **高さ**
- **Spring constant k**
- **Extension or compression x**
- **力**
- **Distance moved**
- **Angle between force and motion**
- **時間**
- **仕事率**
- **Speed at the top**
- **Speed at the bottom**
- **Gravitational acceleration**

## 結果

- Energy or work (J) — 主な結果
- 仕事率 (W)
- Speed at the bottom (m/s)
- 質量 (kg)
- 速さ (m/s)
- 高さ (m)
- Spring constant (N/m)
- Extension (m)
- 力 (N)
- Distance (m)
- 時間 (s)
- Energy or work (kWh)

## 計算式

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

## 計算例

### 1000 kg car at 20 m/s

- Quantity: Kinetic energy ½mv²
- Solve for: エネルギー
- 質量: 1000 kg
- 速さ: 20 m/s
- **Energy or work: 200,000 J**
- **Energy or work: 0.055556 kWh**
- 照合元：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: 速さ
- Energy or work: 100 J
- 質量: 0.145 kg
- **速さ: 37.1391 m/s**
- 照合元：Python 3.8 decimal: v = √(2E/m) = √(200/0.145) = 37.1390676…

### 50 kg lifted 10 m

- Quantity: Gravitational potential energy mgh
- Solve for: エネルギー
- 質量: 50 kg
- 高さ: 10 m
- **Energy or work: 4,903.33 J**
- 照合元：Python 3.8 decimal: 50 × 9.80665 × 10

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

- Quantity: Spring energy ½kx²
- Solve for: エネルギー
- Spring constant k: 200 N/m
- Extension or compression x: 10 cm
- **Energy or work: 1 J**
- 照合元：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
- 力: 100 N
- Distance moved: 5 m
- Angle between force and motion: 60 °
- **Energy or work: 250 J**
- 照合元：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
- 力: 100 N
- Distance moved: 5 m
- Angle between force and motion: 90 °
- **Energy or work: 0 J**
- 照合元：W = Fd cos 90° = 0 (OpenStax UP1 §7.1)

## よくある質問

### 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.

### 「Kinetic and potential energy, work and power calculator」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：8。 例えば、「1000 kg car at 20 m/s」はPython 3.8 decimal: ½ × 1000 × 20² = 200000 J = 0.0555… kWhと照合しています。

### この計算方法の出典は何ですか？

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

## 出典

- [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)