Well, while you're looking thru the sties the above mentioned, I'm just going to add a simpler effort to look at the way you are thinking antennas work...
1/2 Wave antenna?
Divide it by 2 - you get 2 - 1/4 Wave antennas Right?
Where are you feeding the antenna? Where are you attaching your COAX - also known as FEEDLINE
You say why does 1/4 wave need a ground plane - but 1/2 wave doesn't...(well, yes it does, but it's not what you think)
Re-read at the above carefully. You can look at this problem in different ways, change your point of view. How? Start with - It's in how you're feeding the antenna - in a 1/2 wave, it's being 2 - 1/4 wave antennas, you feed it at one end.
Now take a look at your other antenna - your 1/4 wave antenna. It is still the same size - BUT YOU FEED IT AT ONE END, IN THE MIDDLE - You connect one part of your coax wire to it - use the CENTER for that - connected the SHIELD to the other 1/4 wave feed it there THAT THE OTHER HALF STARTS FROM (The Middle) AND GOES IN THE OPPOSITE DIRECTION. Only you don't make the other half, the ground image does. Put the 1/4 wave with the ground image of another 1/4 wave - you make a 1/2 wave LENGTH TOTAL overall antenna correct? Only you feed it at a 1/4 wave point (the middle) - but you make a 1/2 wave antenna you just let the ground image form from the COUNTERPOISE you need to make that 1/4 wave work.
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If you feed a 1/4 wave antenna - you sill need ANOTHER part of it - that missing 1/4 wave is in your ground image...your Counterpoise...or what you use your vehicle - or any other metal you've attached that 1/4 wave antenna to - used as a means to COUNTERPOISE (Offset a lack of) otherwise.
