Every moving thing has kinetic energy, the form of energy that comes with motion. A faster or heavier object has more. The formula has a square in it, so you will need the order of operations: square the speed first, then multiply.
Key idea
$$\text{KE} = \tfrac{1}{2}\,m\,v^2$$ with mass in kg and speed in m/s, giving joules. Because the speed is squared, doubling the speed makes the kinetic energy four times as big.
Why it works
Why squared? To speed something up, you push it through a distance, and a faster object covers more distance during each push. Getting from 0 to 20 m/s takes four times the energy of getting from 0 to 10 m/s, not twice. That is also why a car at highway speed needs about four times the braking distance it needs at half that speed.
Worked example: A rolling ball
Find the kinetic energy of a 2 kg ball rolling at 3 m/s.
- Square the speed first: $3^2 = 9$.
- $\text{KE} = \tfrac{1}{2} \times 2 \times 9 = 9$ J.
Answer: 9 J
Worked example: Double the speed
A 1,000 kg car moves at 20 m/s. Find its kinetic energy, then find it again at 40 m/s.
- At 20 m/s: $\tfrac{1}{2} \times 1{,}000 \times 20^2 = 500 \times 400 = 200{,}000$ J.
- At 40 m/s: $\tfrac{1}{2} \times 1{,}000 \times 40^2 = 500 \times 1{,}600 = 800{,}000$ J.
- Twice the speed gives $2^2 = 4$ times the energy.
Answer: 200,000 J, then 800,000 J
Common mistake
Thinking that doubling the speed doubles the kinetic energy. The speed is squared, so it multiplies the energy by $2^2 = 4$; tripling the speed multiplies it by 9. Check: in your work, the speed should be squared before anything else is multiplied.