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Skill 3 of 13
In Progress

Kinetic energy

An off baseball can knock over a trash can. Clearly, simply by having speed, an object is able to initiate interactions with other objects and make them move. Being in motion is clearly a way to carry energy.

Let’s call this type of energy: ‘dynamic energy’ or ‘motion energy’, or why not simply kinetic energy.1 2 Let’s symbolise it $K$.3

Since there are two types of motion, translation and rotation (the ball can both fly and spin at the same time), we can consider them as causing two types of kinetic energy. It turns out that they can be calculated with these formulas:4

$$K_\text{trans}=\frac12 mv^2\qquad\qquad K_\text{rot}=\frac12 I\omega^2$$

where $m$ and $I$ are mass and moment-of-inertia, and $v$ and $\omega$ are translational and angular speed.5 6 If interested, these formulas are derived in Proof 35.

Wait, is the speed $v$ really squared? Are kinetic energy $K$ and speed $v$ not proportional?7 Does doubling the speed give more than double kinetic energy? (Likewise for $\omega$.)

Yes, indeed. In fact, doubling the speed gives four times the kinetic energy.8

And this is crucial knowledge: Speeding on the motorway with $150\;\mathrm{km/hr}$ instead of keeping the $110\;\mathrm{km/hr}$ limit – increasing the speed with less than half – makes you carry twice as much energy,9 making it much more dangerous than expected.10


  1. Online Etymology Dictionary’ (dictionary), Douglas Harper,
  2. A Course of Lectures on Natural Philosophy and the Mechanical Arts’ (book), Thomas Young, Johnson, vol. 1, 1807,, page 78
  3. Kinetic Energy, Discussion’ (web page), Glenn Elert, The Physics Hypertextbook, (accessed May 23th, 2020)
  4. Kinetic Energy Formula’ (web page),, (accessed Jun. 12th, 2020)
  5. The World of Physics’ (book), John H. Avison, Nelson, 2nd ed., 1989, (accessed Jun. 12th, 2020), ISBN 0-17-438733-4, chapter 8.4, page 148
  6. Sears and Zemansky’s Univesity Physics with Modern Physics’ (book), Hugh D. Young & Roger A. Freedman, Pearson Education, 13th ed., 2012
  7. Kinetic energy’ (encyclopedia), Wikipedia, (accessed Jun. 12th, 2020)
  8. Smart Fortwo Coupe (W450) CDI Specs’ (web page), Ultimate Specs, (accessed Jun. 11th, 2020)