Kinetic Energy with constant forces

The kinetic energy $K$ of a particle with mass $m$ and speed $v$ is somewhat mysteriously defined as

$$K ~=~ {1\over 2}m v^2$$ (1.2)

This is true even in three dimensions but we'll start by considering it in just one dimension. We see now why this is a sensible definition.

If we apply a constant net force $F$ to an otherwise free particle it will accelerate with constant acceleration because $F ~=~ ma$. Remember that with constant acceleration we showed earlier that over some time interval you could relate initial velocity $v_0$, final velocity $v$, and distance traveled $\Delta x$ as

$$v^2 - v_0^2 ~=~ 2a\Delta x$$ (1.3)

Multiply both sides by ${1\over 2}m$, and this becomes $K_f -K_i ~=~ ma \Delta x ~=~ F \Delta x$. But in one dimension $F \Delta x ~=~ W$, the work done by the force F, so we end up with the simple looking formula

$$\Delta K ~=~ W$$ (1.4)

This is called the ``work-energy theorem''. In words this says that the change in kinetic energy is equal to the work done by the force $F$ on mass $m$. So this kinetic energy is a pretty neat thing. If you know its change, you can figure out how much work was done by all the forces acting on the the particle.