Tricky hit

A wooden stick of length \(\ell\) is balanced vertically in a cup on a table. Your job is to hit the stick so that the bottom end of the stick comes out of the cup without touching the cup. If you strike it too low (close to the cup), the bottom of the stick will move forward and hit the wall of the cup. If you strike it too high, the bottom of the stick will move backward and hit the wall of the cup.

What is the distance \(h\) from the top that you should hit the stick so that the bottom of the stick moves vertically upwards, not touching the cup?

Express the distance as \(h=a\ell,\) and give your answer as \(a\) to 3 decimal places.

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