In this range you can get a decent laptop , nothing fancy ...
The key factors will be a decent RAM and processor and latest generation.
With these factors in mind you can find alot of laptops , the battery timing is more or less dependent on the usage ...and majority laptops give you around 3+ hours battery time which decreases with time.
Budget Laptops around $500
The price may vary a bit as newer models hit the market the price decreases of the olders ones, models which are decent enough for a buy
HP Pavilion 15 - P089TX
HP Pavilion 15-AB206TU
Dell Latitude E3550 ci3
HP ProBook - a decent series which ever falls in your range , got for latest 7th gen, even if its i3
Lenovo Ideapad - cheaper but Lenovo is you like as a brand you may opt for it