The Quickest Way Down
Last timeFrom One Particle to a Box of Gas
Energy conservation fixes the speed at each height whatever the path, which turns a question about a shape into a question about a total. The shape that wins is longer than the straight line.
Bend a wire into some shape, thread a bead on it, and let the bead go from the top.
Gravity pulls, the wire guides, nothing rubs. The bead arrives at the bottom end of
the wire after some time, and that time depends on the shape you bent.
Which shape is quickest?
The straight line is the obvious guess and it is wrong, which is the only reason
this question is famous. It is worth being clear about why the guess fails before
anything is calculated. A straight wire is the shortest route, but the bead starts
at rest and a gentle slope keeps it slow for a long while. A wire that drops
steeply at first is longer, and the bead on it is travelling fast very early, and
it then carries that speed along every remaining stretch. One of those effects wins
and it is not the one the straight line is betting on.
Why the question has an answer at all
The reason this can be worked out rather than guessed is the previous lesson. The
wire is a constraint and it does no work, so the bead's energy total contains only
its motion and its height. Set the total at the start equal to the total later and
the speed of the bead depends on how far it has descended and on nothing else at
all.
That is a strong statement. Two beads on completely different wires, passing
through the same height, are going at exactly the same speed. The shape of the wire
has no say in it. So the speed is known everywhere before any shape is chosen,
which is what lets shapes be compared.
- how far the bead has descended from where it started, measured straight down
- its speed there, which depends on h alone. This is the result of the previous lesson doing all the work
- the length of a short piece of the wire, measured along the wire rather than across the gap
- the total journey time, which is the quantity the winning shape makes as small as possible
Three candidates
Here are three wires joining the same two points. All three start at the top left
and finish at the same place lower down and to the right.
The lesson stops here
4 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents