Elastic collisions help

Summarising my working, I said that after the first collision the velocity component parallel to the cushion is unchanged so remains as u cos(20) and the perpendicular component u sin(20) becomes (1/2)u sin(20).

Then in the second collision the direction of the components switch around and so (1/2)u sin(20) remains the same and u cos(20) becomes (2/5)u cos(20).

Then using Pythagoras, the final speed^2 = (1/2)^2 u^2 + (2/5)^2 u^2 = 41/100 u^2 and so the kinetic energy has decreased by 59%.

The answer given in the textbook is 83% and the textbook's method is much more complicated than mine so I'm thinking that my understanding is wrong somewhere.

Also, you seem to ignore the trig part of the components when doing pythagoras at the end?

Thanks I found my mistake.

I stupidly thought about this in my head

(1/2)^2 x sin^2(20) x u^2 + (2/5)^2 cos^2(20) x u^2

and thought the trig disappears due to sin^2(20) + cos^2(20) = 1 but that's clearly wrong. If I use my calculator for the above I get 0.1705...u^2 which gives 83% like the textbook. I thought it was odd that my answer didn't depend on the initial angle!

Here's my diagram:



I didn't really think much about the angles and just thought about how the components change. Can you please give more details about the incorrect assumption I made?

And here's the textbook solution:



Here's the textbook solution: