Conservation Laws

Principles preserving physical quantities in isolated systems. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Conservation laws address one of the deepest questions in physics: what remains invariant when a system changes? Experience shows that certain physical quantities do not change during the evolution of isolated or conservative systems. $E_{mec}$ remains constant when only conservative forces act; total $p$ is conserved in systems with no net external force. These laws are not mere mathematical curiosities: they allow prediction of the final $v$ and reachable $h$ without needing to know the details of the intermediate trajectory.

🟢 Essential level

Imagine a pendulum swinging without friction. At the highest point the body has large $E_{pot}$ and nearly zero $v$; at the lowest point $E_{pot}$ is minimal and $v$ is maximum. Yet $E_{mec}$ is identical at both instants and at every intermediate point. Throughout the full swing, $E_{cin}$ and $E_{pot}$ exchange continuously without their sum ever changing.

The same happens with $p$ in a collision between two billiard balls: each ball changes its $v$ on impact, but the total $p$ of the system remains constant. In both phenomena a quantity stays invariant during the system evolution. Identifying that conserved quantity and understanding why it does not change, despite the complexity of the motion, is the central question of this topic.

A third example appears in a frictionless horizontal spring: when the block passes through the equilibrium point, all the energy is kinetic; at the extremes of its path, all the energy is elastic potential energy. The total energy stays constant at every instant, exactly as in the pendulum, even though the force responsible for the motion is different.

🔵 Formal level

Basic formulas

$E_{mec}$ of a mechanical system is defined as the sum of $E_{cin}$ and gravitational $E_{pot}$:

Total mechanical energy

where $m$ is the mass of the body, $v$ its instantaneous speed, $g$ the gravitational acceleration, and $h$ the vertical distance to the chosen reference level. When the system experiences no non-conservative forces, $E_{mec}$ does not change between any pair of states. This is expressed by the conservation of mechanical energy law:

Conservation of mechanical energy

where subscripts i and f denote the initial and final states. This equation allows calculation of $v$ at any $h$ knowing only the initial values, without needing to know the intermediate path. If friction appears, $E_{mec}$ decreases and the equation must be corrected by including the work of dissipative forces.

Real student questions

Why does the velocity at the bottom not depend on mass? Because both $E_{cin}$ and $E_{pot}$ are proportional to $m$, which cancels when applying the conservation law. $m$ scales the level of $E_{mec}$ but does not determine the final $v$.

Can I use energy conservation if the body rises and falls multiple times? Yes, as long as there are no dissipative forces. $E_{mec}$ is the same at every instant, regardless of how complex the path or how many oscillations occur.

Is linear momentum always conserved? Only in isolated systems, where the net external force is zero. If a wall, the ground, or a rope exerts a force, the $p$ of the body changes. The total $p$ of the complete system (body plus Earth) is always conserved.

Final synthesis

The conservation laws for $E_{mec}$ and $p$ reveal that behind the apparent complexity of motion lie invariant quantities the system keeps intact through $t$. Identifying what is conserved is the first step in any energetic or dynamic problem: conservation does not require the system to be static, only that a physical quantity remains constant throughout the entire evolution.