내다
nae-da
내다 is you pushing what was inside out
The picture Your hand shoving something from inside out into the open. 나다 comes out on its own; 내다 you push. 내다 is you pushing it out by hand; 나다 is it surfacing with nobody pushing at all.
Where the picture shows up in 내다
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push out time you didn't have
시간 좀 내 주세요.
si-gan jom nae ju-se-yo
Please make some time for me.
Two hands pry apart a solid block of a schedule to force one empty square into existence
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push the paper out
숙제 언제까지 내야 돼요?
suk-je eon-je-kka-ji nae-ya dwae-yo
When is the homework due?
A sheet is pulled out of a bag and slid across onto the teacher's desk
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push the money out
오늘은 제가 낼게요.
o-neu-reun je-ga nael-kke-yo
It's on me today.
Notes come out of a wallet and are pushed across the counter
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push the sound out
여기서는 큰 소리 내지 마세요.
yeo-gi-seo-neun keun so-ri nae-ji ma-se-yo
Please don't make loud noises in here.
A palm clamps over a sound that was about to burst out of a mouth, keeping it in
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you make it happen
결국 사고를 냈어요.
gyeol-guk sa-go-reul nae-sseo-yo
In the end I caused an accident.
Your own two hands yank the wheel and manufacture a crash that was never going to happen
By itself: 사고가 났다. Caused by you: 사고를 냈다.
All of it in one picture
내다 is pushing what was inside out into the open, under your own force. Push time and you make time that was not there; push a sheet of paper and the homework is handed in; push money and the bill is paid. Push sound out and you are making a noise; push an event out with your own hands and you have caused an accident. Where 나다 has something rising by itself, 내다 always has a pushing hand somewhere in the frame.