Why Define Kinetic Energy?

When we push or pull an object and change its speed, we do work on it. A natural question arises: where does that work go? To answer this, let's analyze the motion mathematically.

Consider an object of mass $m$ experiencing a net force $F_{\text{net}}$ along its direction of motion. By Newton's second law:

$$F_{\text{net}} = ma$$

For constant acceleration over displacement $d$, the kinematic equation relates initial speed $v_i$, final speed $v_f$, acceleration $a$, and displacement:

$$v_f^2 = v_i^2 + 2ad$$

Solving for acceleration:

$$a = \frac{v_f^2 - v_i^2}{2d}$$

Substitute into Newton's second law and multiply both sides by $d$:

$$F_{\text{net}} d = m \frac{v_f^2 - v_i^2}{2d} d$$ $$F_{\text{net}} d = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$$

The left side is the net work done on the object: $W_{\text{net}} = F_{\text{net}} d$. The right side reveals a quantity that changes when work is done—a quantity that depends on the object's mass and the square of its speed. We call this quantity kinetic energy:

The calculation shows us that the net work done equals the change in this quantity:

This is the work-kinetic energy theorem. Kinetic energy emerges naturally from analyzing how work affects motion. The factor of $\frac{1}{2}$ and the $v^2$ dependence aren't arbitrary—they arise directly from the relationship between force, displacement, and velocity.

This theorem provides a powerful tool: we can find speed changes directly from forces and displacements without tracking acceleration or time.

Example: Box Sliding on a Frictionless Surface

Consider a box of mass $m$ sliding with initial speed $v_i$ on a frictionless surface. A constant force $\vec{F}$ acts parallel to the displacement over distance $d$.