1. Start with Newton's Second Law: $F = ma$. 2. Use the chain rule: $a = v\frac{dv}{dx}$. 3. Substitute: $F = mv\frac{dv}{dx}$. 4. Integrate work: $W = \int F dx = \int mv dv$. 5. Integrate from initial to final velocity to get $W = \Delta KE$.

The work done by a variable force is the area under a force vs. position graph.

Always start with a free-body diagram.

Conservative forces: Work done is path-independent, energy is conserved. Non-conservative forces: Work done is path-dependent, energy is not conserved (e.g., converted to heat).

Work is the result of a force causing displacement. Mathematically, it's $W = Fd\cos\theta$, where $F$ is force, $d$ is displacement, and $\theta$ is the angle between them.

Kinetic Energy is the energy of motion. It's calculated as $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is velocity.

The net work done on an object equals the object’s change in kinetic energy.

Conservative forces (like gravity and springs) depend only on position.

Non-conservative forces (like friction and air resistance) depend on velocity.