Wing tip of the Rutan Voyager?
Tell him what he's won, Johnny!
I still remember watching the takeoff live on TV. The one winglet was badly damaged and sort of flapping around. A chase plane was following and monitoring its progress, and eventually it ripped away. There was some discussion about whether they should abort the flight, but it was decided that it likely didn't compromise the wing integrity, so they flew on.
History!
My train of thought was "why would the museum display something damaged?", as most of the articles there are restored. It had to be something damaged when it did whatever it was famous for. Which made me recall the Voyager damaging it's wingtip taking off for it's historic flight around the globe. Checked to see if it was in the collection, and bingo, there it was, including the photo of it in what appeared to be the same hall as your photo.
There's no Arby's within a hundred miles of me, so you can donate that coupon to the charity of your choice