The work-energy theorem states that the net work done on an object equals its change in kinetic energy. This is not just an abstract relationship. It is a problem-solving powerhouse. Instead of tracking forces, accelerations, and displacements separately, you can connect the initial and final states directly. If you know how much work was done, you know how much the kinetic energy changed, and vice versa.
Work is force times displacement times the cosine of the angle between them. Push a box across the floor with a horizontal force, and the work is simply force times distance. Push at an angle, and only the component of force along the displacement does work. Push perpendicular to the motion, and no work is done. The Moon orbiting Earth experiences a gravitational force, but because the force is perpendicular to the velocity, no work is done and the Moon’s kinetic energy stays constant.
Our Work Calculator handles the angle case. This is important for real scenarios like pushing a lawnmower up a hill, where the force is not aligned with the displacement.
Power is the rate of doing work. A watt is one joule per second. A 100-watt light bulb converts 100 joules of electrical energy to heat and light every second. A car engine producing 150 horsepower delivers about 112,000 watts. The relationship between work, power, and time is straightforward: power equals work divided by time. Our Power Calculator computes any one of these from the other two.
The work-energy theorem is particularly useful for variable forces. If you graph force versus displacement, the area under the curve is the work done. For a spring that follows Hooke’s law (F equals negative kx), the work to compress it by distance x is one half kx squared. This stored energy is called elastic potential energy, and it is why springs bounce back.
Conservative forces like gravity and spring forces have a special property: the work done depends only on the initial and final positions, not on the path taken. This leads to the concept of potential energy. Non-conservative forces like friction do work that depends on the path. The longer the path, the more work friction does. This is why rolling something down a long ramp loses more energy to friction than a short drop.
The theorem applies to everything from billiard balls to spacecraft. When a meteor enters the atmosphere, friction does enormous negative work, converting kinetic energy to heat. When a pole vaulter runs and plants the pole, their kinetic energy converts to elastic potential energy in the bending pole, which then converts to gravitational potential energy at the top of the vault. Energy is conserved throughout, just changing forms.