# Super ball

A friction-less board has the shape of an equilateral triangle of side length $1$ meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards the side $BC$. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is ?

Note by Kudo Shinichi
1 year, 11 months ago

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Consider an array of triangles as shown below, with vertices labelled as shown. The red straight line travels from a vertex A to a vertex B, crossing $9$ edges in the process.

Now fold up the red line, reflecting each portion of the line in the triangle edge it crosses, and we obtain the following track of a ball from vertex A to vertex B, which bounces perfectly off 9 walls.

The original red length moves $5\tfrac12$ m horizontally and $\tfrac12\sqrt{3}$ m vertically, and hence is $\sqrt{31}$ m long. Thus the ball's path is $\sqrt{31}$ m long.

- 1 year, 11 months ago

Amazing solution @Mark Hennings Sir , btw if ball was thrown from a point in the line AB , except the vertices , can then also the ball will land on a vertice after nine times striking the walls? Is it possible ?

- 1 year, 11 months ago

Try playing with a triangular grid and drawing lines. You should be able to convince yourself that you can get from anywhere on an edge to some vertex after any required number of bounces..

- 1 year, 11 months ago

Oh wow , such a beauty of this problem, btw @Mark Hennings Sir , how u think of the idea of making equilateral triangles copies by reflecting many times ?

- 1 year, 11 months ago

Also @Mark Hennings sir is it possible for ball to land at same vertex , from where it started ?

- 1 year, 11 months ago

Fire the ball from one vertex towards the midpoint of the other side, and it will bounce straight back to the vertex it started from. There are other less trivial solutions... Try to find them yourself. I can get from a vertex back to itself after seven bounces, for example.

Coping with bounces by this process of reflection is quite standard. You might have seen it before with a rectangular table, and bouncing a pool ball...

- 1 year, 11 months ago

Got it now @Mark Hennings sir thx a lot.

- 1 year, 11 months ago

You're welcome

- 1 year, 11 months ago

@Steven Chase sir @Mark Hennings sir pls see this problem

- 1 year, 11 months ago