Physics
Work, energy and power calculator
Pick the relation and clear the quantity you want: the formula is rearranged for you. Energy, mass, speed or time — the unknown is simply the field left blank.
Work and energy: one quantity seen from two sides
Work is how energy passes from one body to another: when a force moves something along its own direction, that thing's energy changes by exactly the work done. This is the work-energy theorem, and it explains why work and energy share a unit, the joule. One joule is the work of a force of one newton over one metre.
Kinetic energy is ½mv², and the dependence on the square of the speed is the fact that always surprises: a car at 100 km/h carries four times the energy it had at 50, not twice, and therefore needs four times the braking distance. Gravitational potential energy mgh depends linearly on height, and only on the difference: the zero level can be put wherever it is convenient.
The angle in the work formula is what makes the quantity non-trivial. Only the component of the force parallel to the displacement does work: a perpendicular force, such as the one holding a bag while walking on level ground, does none at all, however tiring it is to sustain. Beyond 90° the cosine turns negative and the work is resistive, removing energy.
Common mistakes
- Dropping the square on the speed in kinetic energy: ½mv is a quantity with no physical meaning, and for v greater than 2 it gives a smaller number than the correct one.
- Using absolute altitude instead of the height difference in potential energy: only the difference between the two points in the problem counts.
- Ignoring the angle in work and always multiplying force by displacement: that is correct only when the force is parallel to the motion, that is when the angle is zero.
Frequently asked questions
What is the formula for kinetic energy?
Eₖ = ½mv², with mass in kilograms and speed in metres per second. A 1200 kg car at 25 m/s has 375,000 joules, that is 375 kJ.
How do you calculate the work done by a force?
W = F·s·cos α, where α is the angle between force and displacement. If the force is parallel to the displacement the cosine is 1 and the work is simply force times displacement; if it is perpendicular the work is zero.
What is the difference between energy and power?
Energy is how much work is done in total, power is how fast. Lifting 100 kg by one metre always takes about 981 joules, whether you do it in a second or a minute; the power, however, goes from 981 watts to 16 watts.
Why is momentum different from kinetic energy?
Momentum mv is a vector and is conserved in every collision; kinetic energy ½mv² is a scalar and is conserved only in elastic ones. Two identical bodies colliding head-on and stopping have conserved momentum, which was zero, but lost all their kinetic energy.
How this calculation works
Kinetic energy: Eₖ = ½·m·v². Gravitational potential energy: Eₚ = m·g·h, with g = 9.81 m/s². Elastic potential energy: Eₑ = ½·k·x², with elastic force F = k·x (Hooke's law). Work: W = F·s·cos α, zero at α = 90°, resistive for α > 90°. Power: P = W/t, with 1 W = 1 J/s and 1 kWh = 3,600,000 J. Momentum: p = m·v. Work-energy theorem: the total work done on a body equals the change in its kinetic energy, W = ΔEₖ.
Related calculators
Motion and projectiles
Uniform and uniformly accelerated motion, free fall and launch at an angle.
Dynamics and Newton's laws
Second law, weight, friction, inclined plane, centripetal force and impulse, with the formula inverting itself.
Collisions and momentum
Head-on collisions, elastic, inelastic and in between: final velocities, energy lost and impulse exchanged.
Rotational dynamics
Moment of inertia, the parallel axis theorem, torque, angular momentum and rolling, with the formula that rearranges itself.