KE = ½mv², solved for the energy, the mass or the speed.
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The Coddy physics course teaches the method: you solve each problem in steps on an interactive board, and it tells you which step went wrong.
Kinetic energy is the energy an object has because it is moving. It depends on two things, the mass m and the speed v, and the formula is KE = ½mv². With m in kilograms and v in metres per second, the answer comes out in joules. A 2 kg ball rolling at 3 m/s carries ½ · 2 · 3² = 9 J.
The speed is squared, and that is the part worth remembering. Double the mass and the energy doubles; double the speed and it goes up four times. It is why braking distance grows so fast with speed: a car at 100 km/h has four times the energy of the same car at 50 km/h, and the brakes have to turn all of it into heat.
The calculator solves the formula for whichever value you leave blank. Give a mass and a speed for the energy, an energy and a mass for the speed, or an energy and a speed for the mass. A speed in km/h or mph is converted to m/s first, as its own step, because squaring a speed in the wrong unit is the most common way to get this wrong.
Grams divide by 1000, pounds multiply by 0.4536. A 145 g baseball is 0.145 kg.
km/h divide by 3.6, mph multiply by 0.44704. 100 km/h is 27.78 m/s.
KE = ½ · m · v². Square first: ½ · 1500 · 27.78² is about 579,000 J, and squaring after multiplying gives a very different number.
Divide by 1000 for kilojoules. 578,704 J is about 579 kJ.
Rounded values from KE = ½mv². The car rows show the speed squared at work: twice the speed, four times the energy.
| Object | Mass | Speed | Kinetic energy |
|---|---|---|---|
| Person walking | 70 kg | 1.4 m/s | 69 J |
| Baseball pitch | 0.145 kg | 40 m/s | 116 J |
| Pistol bullet | 8 g | 360 m/s | 518 J |
| Sprinter | 80 kg | 10 m/s | 4,000 J |
| Car in town | 1,500 kg | 50 km/h | 145 kJ |
| Car on a highway | 1,500 kg | 100 km/h | 579 kJ |
m = 1500 kg, v = 100 km/hConvert the speed first: 100 km/h = 100 · 1000/3600 ≈ 27.778 m/s. Then KE = ½ · 1500 · (100/3.6)² ≈ 578,704 J, about 579 kJ. Putting 100 straight into the formula would give 7,500,000 J, almost 13 times too much.
m = 145 g, v = 40 m/s145 g is 0.145 kg. KE = ½ · 0.145 · 40² = ½ · 0.145 · 1600 = 116 J. Small mass, but the speed is squared.
KE = 500 J, m = 2.5 kgRearrange for v: v = √(2KE/m) = √(2 · 500 / 2.5) = √400 = 20 m/s. Leave the speed box blank and the calculator does exactly these steps.
m = 1500 kg, v = 50 km/h50 km/h is about 13.889 m/s, so KE = ½ · 1500 · (50/3.6)² ≈ 144,676 J. At 100 km/h it was about 578,704 J: four times as much for twice the speed.
KE = ½mv², where m is the mass in kilograms and v the speed in metres per second. The result is in joules. A 2 kg object at 3 m/s has ½ · 2 · 3² = 9 J.
Rearrange the formula: v = √(2KE/m). Multiply the energy by 2, divide by the mass, then take the square root. 500 J on a 2.5 kg object gives √400 = 20 m/s.
Rearrange for m: m = 2KE/v². An object with 900 J moving at 30 m/s has a mass of 2 · 900 / 900 = 2 kg.
The joule, J. One joule is one kilogram times one metre squared per second squared. Large values are given in kilojoules (1 kJ = 1000 J); food energy uses kilocalories, 1 kcal = 4184 J.
It goes up four times, because the speed is squared. Three times the speed gives nine times the energy. Doubling the mass, by contrast, only doubles it.
No. The mass is positive and the speed is squared, so ½mv² is zero for an object at rest and positive for anything moving, whichever way it goes.
Kinetic energy is energy of motion, ½mv². Potential energy is stored energy of position, mgh for a height above the ground. A falling ball turns one into the other: at the top it is all potential, just before landing almost all kinetic.
