You can use javascript/jquery to do what you want. Simply add the submit event on the form and prevent default. In that event you get the search value and send the user to the new page search/hello. Let me know if you need help gretting started with the code.
Notice the added id="searchForm"
<form id="searchForm" action="http://www.mywebsite.com/search/" method="get"><input class="nice_search" type="text" name="q" placeholder="Search...">
<input class="nice_submit" type="submit" value="SEARCH"></form>
Something like this
$('#searchForm').submit(function(e){
e.preventDefault();
var search = $('.nice_search').val();
var url = $(this).attr('action');
//alert('search='+search+ ' url='+url);
window.location.href = url+search;
});