How to make emacs clear search when pressing Ctrl+g?

[Joseph Garvin]

Sometimes when I'm doing a forward search (Ctrl+s) I realize I
actually wanted to search for something different. As a shortcut to
clear the search rather than having to backspace, I press Ctrl+g to
cancel the search then press Ctrl+s again to begin a new search. But
if you do this without moving the cursor between the Ctrl+g and the
Ctrl+s steps, then the minibuffer will keep what I was searching for
before in it, defeating the whole point. Is there a good reason this
is the default behavior? I'm not sure what the advantage would be. And
regardless, is there an easy way to make it always clear as if I had
moved the cursor between the Ctrl+g and Ctrl+s steps?