Elastic collision in 1D

In a one-dimensional elastic collision both momentum and kinetic energy are conserved. These are two equations, and with two unknowns (the final velocities $v_1'$ and $v_2'$) the problem is fully determined.

Elastic collision in 1D: the balls (partly) exchange velocities. The lighter body tends to bounce back.

The two conservation laws are written:

$\begin{aligned} m_1 v_1 + m_2 v_2 &= m_1 v_1' + m_2 v_2' \\ \tfrac{1}{2}m_1 v_1^2 + \tfrac{1}{2}m_2 v_2^2 &= \tfrac{1}{2}m_1 v_1'^2 + \tfrac{1}{2}m_2 v_2'^2 \end{aligned}$

The second equation is quadratic and awkward to solve directly. There is, however, a trick that makes it linear.

Principle — Elastic-collision trick

Conservation of kinetic energy, combined with conservation of momentum, can be rewritten as $\ev{v_{\text{rel,prima}} = -\,v_{\text{rel,dopo}}}$ that is $v_1 - v_2 = -(v_1' - v_2')$: the relative velocity of the two bodies reverses sign.

The trick in practice

Instead of the pair (conservation of $p$) + (conservation of $E_\text{cin}$, quadratic), one uses the much more manageable pair: $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$ $v_1 - v_2 = v_2' - v_1'$ Two linear equations in the unknowns $v_1'$, $v_2'$: solved in a few steps.

The reversal of the relative velocity also tells us something physically important: the two bodies move apart after the collision at the same speed with which they were approaching before. This is the restitution coefficient $e = 1$ of the perfectly elastic collision. From this it follows, for example, that a very light ball hitting a stationary wall bounces back at the same speed with which it arrived.

The full numerical calculation is in Esercizio svolto — urto elastico 1D; the general formula for a stationary target is proved in Dimostrazione urto elastico 1D.

Links

Topics: Quantità di moto e urti Concepts: Urto elastico · Conservazione della quantità di moto · Energia cinetica Skills: Impostazione simbolica

Related exercises: Urto elastico tra due biglie · Dimostrazione urto elastico 1D · La culla di Newton