Master kinetic energy with formulas and examples. Step-by-step calculations for KE = ½mv², work-energy theorem, rotational KE, and practice problems. Free classroom resource.
Ever wanted to put a number on motion? That’s what kinetic energy formulas do. The main formula is KE = ½mv², but there’s a whole family of related formulas. They help you calculate everything from a rolling ball to a spinning planet. This page gives you the formulas, step-by-step methods, and plenty of practice.
| Formula | What It Calculates |
|---|---|
| KE = ½mv² | Translational (straight-line) kinetic energy |
| W = ΔKE | Work-energy theorem |
| KE = ½Iω² | Rotational kinetic energy |
| KE total = KE trans + KE rot | Total KE of a rolling object |
Let’s break down KE = ½ × m × v² piece by piece.
½ (one-half): Comes from the math of acceleration. The average velocity during acceleration from rest is half the final velocity. m (mass): Kilograms. Direct relationship — double the mass, double the KE. v² (velocity squared): This is the big one. Small speed changes produce big energy changes.
Here’s how you calculate kinetic energy every time:
Example: A 12 kg dog sprints at 4 m/s. KE = ½ × 12 × 4² = ½ × 12 × 16 = ½ × 192 = 96 J. That’s enough energy to knock you off your feet.
Watch out for these:
The work-energy theorem connects kinetic energy to work: W = ΔKE = ½mv² - ½mu².
Here’s the simple idea: positive work (pushing to speed up) increases KE. Negative work (friction braking) decreases it. The total work equals the total change in KE. Think of work as the “currency” you use to change kinetic energy.
A 1,200 kg car slows from 20 m/s to 10 m/s. How much work do the brakes do?
KE_initial = ½ × 1,200 × 20² = 240,000 J KE_final = ½ × 1,200 × 10² = 60,000 J W = 60,000 - 240,000 = -180,000 J
The negative sign means the brakes removed energy - all 180,000 J turned into heat.
Spinning objects have their own kind of kinetic energy: KE_rot = ½ I ω².
Translational KE cares about mass and straight-line speed. Rotational KE cares about how the mass is distributed and how fast it spins.
Ever seen a figure skater pull their arms in and spin faster? That’s rotational KE in action. Pulling their arms in decreases their moment of inertia. The rotational KE stays nearly the same, but the angular velocity jumps.
A ball rolling without slipping has both types: KE_total = ½mv² + ½Iω².
For a solid sphere, about 71% of the KE is translational and 29% rotational. Same proportions for a marble or a bowling ball.
Kinetic energy is how much “oomph” a moving thing has. The heavier it is and the faster it goes, the more oomph.
A tennis ball rolling slowly has a tiny bit of oomph. Throw it hard, and it has way more. A bowling ball rolling at the same speed as the tennis ball has much, much more.
Try catching a gently tossed tennis ball, then one thrown hard. The hard one stings — that’s the extra kinetic energy. You just did a science experiment with your own hands!
Line up three paper cups. Roll a marble slowly toward them. How many fall? Now roll it faster. How many now? The faster marble has more KE, so it knocks over more cups. That’s the formula in action. See? You already understand kinetic energy.
You’re ready for real calculations. Here are practice problems with full step-by-step solutions.
A 45 kg student runs at 6 m/s during a race. Calculate their kinetic energy.
Solution: KE = ½ × 45 × 6² KE = ½ × 45 × 36 KE = ½ × 1,620 KE = 810 J
That’s 810 joules - enough energy to lift the student about 1.8 meters straight up.
A 2,500 kg delivery truck travels at 15 m/s (about 54 km/h). What is its kinetic energy?
Solution: KE = ½ × 2,500 × 15² KE = ½ × 2,500 × 225 KE = ½ × 562,500 KE = 281,250 J
That’s the energy equivalent of about 67 grams of TNT. No wonder truck crashes are so destructive.
Object A has mass 4 kg and velocity 2 m/s. Object B has mass 2 kg and velocity 4 m/s. Which has more kinetic energy?
Object A: KE = ½ × 4 × 2² = ½ × 4 × 4 = 8 J
Object B: KE = ½ × 2 × 4² = ½ × 2 × 16 = 16 J
Answer: Object B has twice the kinetic energy, even though it has half the mass. The higher speed (squared) more than makes up for the lower mass.
A 70 kg cyclist accelerates from 2 m/s to 8 m/s. How much work did they do?
Solution: KE_initial = ½ × 70 × 2² = ½ × 70 × 4 = 140 J KE_final = ½ × 70 × 8² = ½ × 70 × 64 = 2,240 J W = 2,240 - 140 = 2,100 J
The cyclist did 2,100 joules of work to speed up. That energy came from the chemical energy in their muscles.
A merry-go-round with moment of inertia 250 kg·m² spins at 3 rad/s. Find its rotational KE.
Solution: KE_rot = ½ × 250 × 3² = ½ × 250 × 9 = 1,125 J
A 1,500 kg car at 30 m/s (108 km/h): KE = ½ × 1,500 × 30² = 675,000 J. The same car at 15 m/s: 168,750 J. The car at 108 km/h has four times the KE - why speed limits save lives.
A 0.145 kg baseball at 45 m/s (major league fastball): KE = ½ × 0.145 × 45² = 146.8 J hitting the catcher’s mitt in milliseconds.
Give students five objects with known masses and a way to measure speed. Have them calculate KE for each and rank them. A fast tennis ball can beat a slow basketball!
Last updated: June 15, 2026
What is the formula for kinetic energy?
A 4 kg object moves at 3 m/s. What is its kinetic energy?
What does the work-energy theorem state?
Rotational kinetic energy depends on which two factors?
A bowling ball and a tennis ball roll at the same speed. Which has more translational kinetic energy?
Answers: B: KE = ½mv², C: 18 J, B: Work equals change in kinetic energy, B: Moment of inertia and angular velocity, B: The bowling ball
What is the basic kinetic energy formula?
The basic formula is KE = ½mv². Kinetic energy equals one-half times mass times velocity squared. The result is in joules (J).
How do you calculate kinetic energy step by step?
Square the velocity, multiply by the mass, then divide by 2. For a 10 kg object moving at 5 m/s: 5² = 25, × 10 = 250, ÷ 2 = 125 J.
What is the work-energy theorem?
The work-energy theorem says the net work done on an object equals its change in kinetic energy. W = ΔKE = ½mv² – ½mu².
What is rotational kinetic energy?
Rotational kinetic energy is the energy of a spinning object. The formula is KE = ½Iω², where I is moment of inertia and ω is angular velocity.
How do you solve kinetic energy problems?
Identify what you know (mass, velocity, initial KE), write the formula, substitute values with units, calculate carefully, and check your answer.
What is the difference between translational and rotational kinetic energy?
Translational KE is energy of straight-line motion (KE = ½mv²). Rotational KE is energy of spinning motion (KE = ½Iω²). An object can have both at once.
Why does the formula have a ½ in it?
The ½ comes from integrating force over distance using Newton's second law. It's not arbitrary - it falls out of the derivation from work and acceleration.
