The two parallel tables

In collision and explosion problems the most systematic and reliable method is to fill in two parallel tables: one for momentum $\vv{p}$ and one for energy $E$. The two tables are solved simultaneously and together provide enough equations to find all the unknowns.

Principle — Two-table method

1. Table of $\vv{p}$: one column for each body, plus the column $\vv{p}_\text{tot}$. If there are no impulsive external forces, $\vv{p}_\text{tot}$ is the same in the before row (A) and the after row (B).
2. Table of $E$: one column for each type of energy ($E_{\text{cin},1}$, $E_{\text{cin},2}$, and then $E_\text{scoppio}$ or $E_\text{term}$), plus the column $E_\text{tot}$, which is conserved.
3. Solve the system formed by the two tables.

The two parallel tables: $\vv{p}$ is conserved (vector, up to 2 equations in 2D), $E$ is conserved (scalar, 1 equation). Together they give enough equations to solve the problem.

The advantage of the method is that it makes it impossible to “forget” an equation: row A and row B of the total column must match, and this is the conservation condition. The column "$E_?$" holds the energy that is not kinetic — the $E_\text{scoppio}$ released in an explosion, or the $E_\text{term}$ produced in an inelastic collision — and lets explosions, elastic collisions and inelastic collisions be treated with the same scaffolding.

The full numerical application is in Esercizio svolto — urto parzialmente anelastico.

Links

Topics: Quantità di moto e urti Concepts: Conservazione della quantità di moto · Energia cinetica Skills: Metodo delle due tabelle Methods: Metodo delle due tabelle parallele

Related exercises: Esercizio svolto — esplosione in due frammenti · Urto elastico tra due biglie · Pendolo balistico (altezza)