Work and kinetic energy

Forces explain motion instant by instant. Energy takes a different deal: give up the play-by-play, and in exchange get answers that skip straight to the ending. This lesson builds the currency that deal is priced in.

Work: force through a distance

A force does work when it pushes something through a distance:

$W = F\,d\cos\theta$

The cos θ keeps the accounting honest — only the part of the force along the motion counts:

• Push a cart forward (θ = 0): full credit, W = Fd.
• Carry a bag horizontally: gravity pulls down, motion is sideways (θ = 90°), so gravity does zero work — however heavy the bag feels.
• Friction on a sliding box (θ = 180°): negative work — it drains energy.

Units: newton × meter = joule (J). Lifting an apple one meter ≈ 1 J.

Kinetic energy

Push an object from rest with constant force and combine two things you already own — F = ma and v² = 2aΔx. The work done comes out as:

$W = F d = m a \cdot \frac{v^2}{2a} = \tfrac{1}{2}mv^2$

That quantity is the kinetic energy: the work it took to bring the object to speed v, and exactly what it can pay back when something stops it.

The v is squared, and that has consequences you can feel: at double the speed a car carries four times the kinetic energy — which is why its braking distance quadruples, not doubles.

The work–energy theorem

$W_{net} = \Delta KE$

Net work equals the change in kinetic energy. Positive net work speeds things up; negative slows them down; zero net work (the cruising car, the orbiting satellite) leaves speed untouched.

It's not a new law — it's F = ma with both sides multiplied by distance. But it trades vectors and time for a single scalar bookkeeping line, and that trade wins whenever the question starts with "how fast" instead of "when."

Why this matters

Work is how energy enters and leaves an object. Next lesson adds a storage account — potential energy — and with it the most useful shortcut in mechanics: conservation.