Calculator of Alexandria

Kinetic energy calculator

Calculate kinetic energy from mass and speed with KE = ½mv², or solve for the mass or speed.

Solve for

Formula

Ek=12mv2E_k = \tfrac{1}{2} m v^2
How it's derived

The formula

Ek=12mv2E_k = \tfrac{1}{2} m v^2

EkE_k is kinetic energy in joules, mm mass in kilograms and vv speed in metres per second.

Where it comes from

Kinetic energy is the work needed to bring an object from rest up to speed vv. Work is force times distance, W=F dW = F\,d.

Push a mass mm with a constant force FF. Newton’s second law gives it a constant acceleration a=F/ma = F/m. For constant acceleration from rest, speed and distance are linked by

v2=2ad⇒d=v22av^2 = 2 a d \quad\Rightarrow\quad d = \frac{v^2}{2a}

So the work done is

W=Fd=(ma)⋅v22a=12mv2W = F d = (m a) \cdot \frac{v^2}{2a} = \tfrac{1}{2} m v^2

The acceleration cancels: however you get to speed vv, it takes the same energy. With calculus the same result holds for any changing force, since ∫F dx=∫mdvdt dx=∫mv dv=12mv2\int F\,dx = \int m \frac{dv}{dt} \, dx = \int m v \, dv = \tfrac{1}{2}mv^2.

Worked example

A 1,500 kg car at 50 km/h. First convert: 50/3.6=13.950 / 3.6 = 13.9 m/s.

Ek=12×1500×13.92=144,700 J≈145 kJE_k = \tfrac{1}{2} \times 1500 \times 13.9^2 = 144{,}700\ \text{J} \approx 145\ \text{kJ}

Watch out for

  • Speed counts twice. Energy grows with the square of speed, so at 100 km/h the same car carries four times the energy, and needs roughly four times the braking distance.
  • Convert to m/s before using the formula by hand. The calculator does it for you.

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