Newton's cradle

Problem

You know Newton’s cradle, with five spheres in a row? You lift a single sphere on one side and let it drop: on the other side exactly one sphere flies off, while the three in the middle stay still. Why? Show that any other configuration violates conservation of kinetic energy.

Solution

The spheres all have the same mass $m$, and collisions between steel spheres are practically elastic: both total momentum and total kinetic energy are conserved. The launched sphere arrives with speed $v$. Suppose that on the other side $n$ spheres leave, all with the same speed $v'$ (the remaining ones stay still). Conservation of momentum: $m\,v = n\,m\,v' \;\Rightarrow\; v' = \frac{v}{n}$ Conservation of kinetic energy: $\tfrac{1}{2}\,m\,v^2 = \tfrac{1}{2}\,n\,m\,v'^2$ Substituting $v'=v/n$: $\tfrac{1}{2}\,m\,v^2 = \tfrac{1}{2}\,n\,m\,\frac{v^2}{n^2} = \tfrac{1}{2}\,m\,\frac{v^2}{n} \;\Rightarrow\; 1 = \frac{1}{n} \;\Rightarrow\; n = 1$ So exactly one sphere leaves, with $v'=v$. If two left ($n=2$, each at $v/2$), momentum would still balance ($2\cdot m\,\tfrac{v}{2}=mv$), but the final kinetic energy would be $\tfrac{1}{2}\,m\,\tfrac{v^2}{2}$, i.e. half: energy would be missing, impossible in an elastic collision. The only configuration that conserves both $\vv{p}$ and $E_\text{kin}$ together is: one goes in, one comes out. $\ev{n=1:\ \text{exactly one sphere leaves at speed } v}$

Links

Topics: Momentum and collisions Concepts: Elastic collision · Conservation of momentum · Kinetic energy Skills: Symbolic setup Objects: Newton’s cradle