Derive the equation for kinetic energy, KE = 1/2mv2, using Newton's laws and kinematic equations.

Work done ($W$) by a constant force ($F$) over a displacement ($S$) is given by:

$$W = F \cdot S$$

According to Newton's second law, force ($F$) is the product of mass ($m$) and acceleration ($a$):

$$F = ma$$

For an object accelerating from an initial velocity $u$ to a final velocity $v$ over a displacement $S$ with constant acceleration $a$, the following kinematic equation applies:

$$v^2 = u^2 + 2aS$$

Rearranging this equation to solve for $aS$:

$$2aS = v^2 - u^2$$ $$aS = \frac{v^2 - u^2}{2}$$

$$W = m \left(\frac{v^2 - u^2}{2}\right)$$ $$W = \frac{1}{2}m(v^2 - u^2)$$

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. If an object starts from rest, its initial velocity $u = 0$. The work done to accelerate it to a velocity $v$ is its kinetic energy ($KE$).
Setting $u = 0$:

$$W = \frac{1}{2}m(v^2 - 0^2)$$ $$W = \frac{1}{2}mv^2$$

Since this work done is stored as kinetic energy:

$$KE = \frac{1}{2}mv^2$$