\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Chapter 6: Energy
9.

Collisions

We saw in the previous chapter that the total momentum of two colliding objects is the same before and after the collision. This is not true for energy: the fact that you can hear a collision means that some of the energy from the two objects has been released into the environment as sound energy, and more has been converted into thermal energy, warming up the objects and their environment.

A collision where no energy is lost at all is called an elastic collision. Here's one example where the total kinetic energy before and after the collision is $14\u{J}$:

Column

initial

(no alternate text)
$\mathbf{\RED{1kg}}$ $\mathbf{\BLUE{3kg}}$ Total
Momentum $(1\u{kg})(+4\u{m/s})$ $(3\u{kg})(-2\u{m/s})$ $\mathbf{-2Ns}$
Kinetic
Energy 		$\frac12(1\u{kg})(+4\u{m/s})^2$ $\frac12(3\u{kg})(-2\u{m/s})^2$ $\mathbf{14J}$

Column

final

(no alternate text)
$\mathbf{\RED{1kg}}$ $\mathbf{\BLUE{3kg}}$ Total
Momentum $(1\u{kg})(-5\u{m/s})$ $(3\u{kg})(+1\u{m/s})$ $\mathbf{-2Ns}$
Kinetic
Energy 		$\frac12(1\u{kg})(-5\u{m/s})^2$ $\frac12(3\u{kg})(+1\u{m/s})^2$ $\mathbf{14J}$

A collision which loses some energy to the environment is called an inelastic collision. The collision which loses the most amount of energy is one where the two objects collide and stick together, moving at the same speed afterward; we call this a maximally inelastic collision.

Column

initial

(no alternate text)
$\mathbf{\RED{1kg}}$ $\mathbf{\BLUE{3kg}}$ Total
Momentum $(1\u{kg})(+4\u{m/s})$ $(3\u{kg})(-2\u{m/s})$ $\mathbf{-2Ns}$
Kinetic
Energy 		$\frac12(1\u{kg})(+4\u{m/s})^2$ $\frac12(3\u{kg})(-2\u{m/s})^2$ $\mathbf{14J}$

Column

final

(no alternate text)
$\mathbf{\RED{1kg}}$ $\mathbf{\BLUE{3kg}}$ Total
Momentum $(1\u{kg})(-\frac12\u{m/s})$ $(3\u{kg})(-\frac12\u{m/s})$ $\mathbf{-2Ns}$
Kinetic
Energy 		$\frac12(1\u{kg})(-\frac12\u{m/s})^2$ $\frac12(3\u{kg})(-\frac12\u{m/s})^2$ $\mathbf{0.5J}$

In this example, the kinetic energy drops from 14.0J to 0.5J, and so 13.5J of energy is lost to sound or heat when the two balls stick together.