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 Azimuddin Sheikh
3 months, 2 weeks 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 99 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 5125\tfrac12 m horizontally and 123\tfrac12\sqrt{3} m vertically, and hence is 31\sqrt{31} m long. Thus the ball's path is 31\sqrt{31} m long.

Mark Hennings - 3 months, 2 weeks ago

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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 ?

Azimuddin Sheikh - 3 months, 2 weeks ago

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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..

Mark Hennings - 3 months, 2 weeks ago

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@Mark Hennings 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 ?

Azimuddin Sheikh - 3 months, 2 weeks ago

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

Azimuddin Sheikh - 3 months, 2 weeks ago

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@Azimuddin Sheikh 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...

Mark Hennings - 3 months, 2 weeks ago

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@Mark Hennings Got it now @Mark Hennings sir thx a lot.

Azimuddin Sheikh - 3 months, 2 weeks ago

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@Azimuddin Sheikh You're welcome

Mark Hennings - 3 months, 2 weeks ago

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

Azimuddin Sheikh - 3 months, 2 weeks ago

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