Physical Problem - triangle as a pool table

[2013]

Physical Problem -- triangle as a pool table

Hello,
can you please help me?

It is an irregular triangle as a pool table:
At point A, there is a small billiard ball.
How should the bullet hit, that it hits first the band 1, then 2, and finally with the band 3, after a shock again arrives at point A?

incidence angle = failure angle

Please help me, thanks in advance. :)

 

[haruspex]

Start by drawing a possible trajectory in the triangle. Fill in unknowns for distances and angles, and see what equations you can write relating them.

 

[2013]

I have drawn angle bisects and construct a possibility.

But how can I set the angles in a relationship?
How can I calculate a possible solution?

 

[gneill]

I notice that there are no hard numbers given for the triangle dimensions or the position of the projectile/target. Are we to assume that you are looking to find a geometrical approach? If so, imagine that you were to replace the pool table sides with mirrored walls.

From any point on the table your line of site would hit a mirror and you'd see a reflection of yourself and the table. In fact, since light follows the same sort of trajectory as a projectile does with regards to angles of incidence and angles of reflection, a shot along a sight line will follow the same path that you "see" along that sight line, effectively running a straight line "through" the reflection.

So, on wall 1 construct a triangle that is a reflection of the original triangle (sitting "behind" the wall as seen from inside the original triangle). The trajectory of a shot aimed at that wall will have a reflection that is a straight-line continuation of the projectile's approach trajectory. Eventually this trajectory will reach wall 2 of the new, reflected triangle. Can you see how to continue this process until the reflection trajectories have "passed through" all three walls? You should be able to "see" the eventual target along a straight-line path of the reflections...

 

[2013]

I have reflected the triangle on all three sides.
Can you please show me what I have to do now?

Thank you, for your help :)

 

[gneill]

I have reflected the triangle on all three sides.
Can you please show me what I have to do now?

Thank you, for your help :)

Consider only the triangle that you built out from side one. Take THAT triangle and build its reflection out from its side 2. Do the same for this new triangle, building out its reflection from its side three. Locate the "target" appropriately in this last triangle. Now a straight line drawn through these triangles represents an "unfolding" (or technically, an "evolution") of a trajectory in the original triangle that reflects off of its sides in the order 1, 2, 3.

 

[2013]

I hope I have reflected it, just like you meant it.
But I don´t know what I have to do now.

Could you please help me?
Is it right, what I have done?

 

[gneill]

You've got the right idea now, but your triangles are getting distorted; they should be similar triangles which have the same side lengths as the original and are only rotated/reflected versions of the original.

 

[2013]

yes, I just could not draw differently on the computer

But how should I draw the straight line through these triangles?
Where is the target in the large triangle?

 

[gneill]

yes, I just could not draw differently on the computer

Okay, but you'll have to work with an accurate diagram to get good results.

But how should I draw the straight line through these triangles?

From the initial position in the original to the target in the last triangle.

Where is the target in the large triangle?

It's in the same relative position as the initial position in the original triangle. The initial position gets reflected, maintaining the geometry, through the successive triangles.

 

[2013]

Is that right up to the point?
What is the next step?
Do I have to fold the triangles back mentally and transfer the individual lines in the original triangle?

 

[gneill]

Is that right up to the point?
What is the next step?
Do I have to fold the triangles back mentally and transfer the individual lines in the original triangle?

Yes, that's the idea. You may find it convenient to locate the points of intersection along the walls for each of the crossings from one triangle to next, then transfer those locations to the walls of the original triangle. That's where the projectile will hit each wall as it travels.

Here's my version of your diagram that preserves the triangle dimensions: