Kinetic Energy Calculator (KE = ½mv²)
Solve the kinetic energy formula KE = ½mv² for energy, mass, or velocity. Enter any two values and get the third with step-by-step working shown.
The kinetic energy formula
Kinetic energy is the energy an object has because it is moving:
- KE — kinetic energy, in joules (J)
- m — mass, in kilograms (kg) — always positive
- v — velocity, in metres per second (m/s) — can be negative (direction), but only its square matters here
Enter any two values and leave the third blank; the calculator solves for it and shows the substitution step.
Worked example
A 10 kg object moves at 3 m/s. What is its kinetic energy?
- Formula: KE = ½mv²
- Substitute: KE = ½ × 10 × 3²
- Result: KE = 45 J
Solving in reverse
The same formula rearranges to find mass or velocity when kinetic energy is known:
- Mass: m = 2·KE / v² — enter KE and v, leave m blank.
- Velocity: v = √(2·KE / m) — enter KE and m, leave v blank (a square root, so the result is always the non-negative speed).
For example, 45 J of kinetic energy at 3 m/s means m = 2 × 45 / 9 = 10 kg. And 45 J of kinetic energy from a 10 kg mass means v = √(2 × 45 / 10) = 3 m/s.
Common mix-ups
- Forgetting the ½. KE = mv² (without the one-half) is a common slip that doubles the answer.
- Forgetting to square v. Kinetic energy is proportional to velocity squared, not velocity itself.
- Entering a negative kinetic energy when solving for velocity. Because velocity is found through a square root, a negative KE has no real solution — check your input if the calculator reports this.
Where this shows up
Kinetic energy connects directly to work through the work-energy theorem: the net work done on an object equals its change in kinetic energy. It is also the energy counterpart to momentum — both depend on mass and velocity, but momentum scales linearly with v while kinetic energy scales with v².
Frequently asked questions
- What is the formula for kinetic energy?
- Kinetic energy is KE = ½mv², where KE is energy in joules (J), m is mass in kilograms (kg), and v is velocity in metres per second (m/s).
- How do I find velocity from kinetic energy?
- Rearrange the formula to v = √(2·KE / m). Enter the kinetic energy and mass, leave velocity blank, and the calculator solves it — including taking the square root — for you.
- Why does doubling speed quadruple the kinetic energy?
- Because velocity is squared in the formula. Doubling v multiplies v² by 4, so KE also multiplies by 4 — this is why high-speed collisions are so much more damaging than slow ones.
- Can kinetic energy be negative?
- No. Kinetic energy depends on v², which is always zero or positive, and mass must be positive, so kinetic energy itself can never be negative. If you enter a negative energy while solving for velocity, there is no real answer.
- What happens if the numbers I enter give no real solution?
- Solving for velocity needs a square root of 2·KE/m. If you enter a negative kinetic energy, that quantity is negative and has no real square root, so the calculator reports that no real solution exists.
- Why must mass be positive?
- Mass represents a physical amount of matter, which cannot be zero or negative. A zero mass would make the KE = ½mv² formula meaningless for finding velocity (division by zero).
- What units does this calculator use?
- SI units — joules (J) for energy, kilograms (kg) for mass, and metres per second (m/s) for velocity. Convert other units (pounds, mph, calories) to SI before entering them.