---
title: "Elastic collision"
date: 2021-10-04
categories:
- Physics
- Classical mechanics
layout: "concept"
---

In an **elastic collision**,
the sum of the colliding objects' kinetic energies
is the same before and after the collision.
In contrast, in an **inelastic collision**,
some of that energy is converted into another form,
for example heat.


## One dimension

In 1D, not only the kinetic energy is conserved, but also the total momentum.
Let $v_1$ and $v_2$ be the initial velocities of objects 1 and 2,
and $v_1'$ and $v_2'$ their velocities afterwards:

$$\begin{aligned}
    \begin{cases}
        \quad\! m_1 v_1 +\:\:\: m_2 v_2
        = \quad m_1 v_1' +\:\:\: m_2 v_2'
        \\
        \displaystyle\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2
        = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2
    \end{cases}
\end{aligned}$$

After some rearranging,
these two equations can be written as follows:

$$\begin{aligned}
    \begin{cases}
        m_1 (v_1 - v_1')
        \qquad\quad\:\;\; = m_2 (v_2' - v_2)
        \\
        m_1 (v_1 - v_1') (v_1 + v_1')
        = m_2 (v_2' - v_2) (v_2 + v_2')
    \end{cases}
\end{aligned}$$

Using the first equation to replace $m_1 (v_1 \!-\! v_1')$
with $m_2 (v_2 \!-\! v_2')$ in the second:

$$\begin{aligned}
    m_2 (v_1 + v_1') (v_2' - v_2)
    = m_2 (v_2 + v_2') (v_2' - v_2)
\end{aligned}$$

Dividing out the common factors
then leads us to a simplified system of equations:

$$\begin{aligned}
    \begin{cases}
        \qquad\;\; v_1 + v_1'
        = v_2 + v_2'
        \\
        m_1 v_1 + m_2 v_2
        = m_1 v_1' + m_2 v_2'
    \end{cases}
\end{aligned}$$

Note that the first relation is equivalent to $v_1 - v_2 = v_2' - v_1'$,
meaning that the objects' relative velocity
is reversed by the collision.
Moving on, we replace $v_1'$ in the second equation:

$$\begin{aligned}
    m_1 v_1 + m_2 v_2
    &= m_1 (v_2 + v_2' - v_1) + m_2 v_2'
    \\
    (m_1 + m_2) v_2'
    &= 2 m_1 v_1 + (m_2 - m_1) v_2
\end{aligned}$$

Dividing by $m_1 + m_2$,
and going through the same process for $v_1'$,
we arrive at:

$$\begin{aligned}
    \boxed{
        \begin{aligned}
            v_1'
            &= \frac{(m_1 - m_2) v_1 + 2 m_2 v_2}{m_1 + m_2}
            \\
            v_2'
            &= \frac{2 m_1 v_1 + (m_2 - m_1) v_2}{m_1 + m_2}
        \end{aligned}
    }
\end{aligned}$$

To analyze this result,
for practicality, we simplify it by setting $v_2 = 0$.
In that case:

$$\begin{aligned}
    v_1'
    = \frac{(m_1 - m_2) v_1}{m_1 + m_2}
    \qquad \quad
    v_2'
    = \frac{2 m_1 v_1}{m_1 + m_2}
\end{aligned}$$

How much of its energy and momentum does object 1 transfer to object 2?
The following ratios compare $v_1$ and $v_2'$ to quantify the transfer:

$$\begin{aligned}
    \frac{m_2 v_2'}{m_1 v_1}
    = \frac{2 m_2}{m_1 + m_2}
    \qquad \quad
    \frac{m_2 v_2'^2}{m_1 v_1^2}
    = \frac{4 m_1 m_2}{(m_1 + m_2)^2}
\end{aligned}$$

If $m_1 = m_2$, both ratios reduce to $1$,
meaning that all energy and momentum is transferred,
and object 1 is at rest after the collision.
Newton's cradle is an example of this.

If $m_1 \ll m_2$, object 1 simply bounces off object 2,
barely transferring any energy.
Object 2 ends up with twice object 1's momentum,
but $v_2'$ is very small and thus negligible:

$$\begin{aligned}
    \frac{m_2 v_2'}{m_1 v_1}
    \approx 2
    \qquad \quad
    \frac{m_2 v_2'^2}{m_1 v_1^2}
    \approx \frac{4 m_1}{m_2}
\end{aligned}$$

If $m_1 \gg m_2$, object 1 barely notices the collision,
so not much is transferred to object 2:

$$\begin{aligned}
    \frac{m_2 v_2'}{m_1 v_1}
    \approx \frac{2 m_2}{m_1}
    \qquad \quad
    \frac{m_2 v_2'^2}{m_1 v_1^2}
    \approx \frac{4 m_2}{m_1}
\end{aligned}$$



## References
1.  M. Salewski, A.H. Nielsen,
    *Plasma physics: lecture notes*,
    2021, unpublished.