I have a project for which I need to cut a groove in 44x44mm wooden plank. The groove is supposed to be 15mm deep and about 3-4mm wide, as well as no less than 150mm long. It is supposed to be positioned in the middle of one of the sides, aligned in the centre of the plank's face. The purpose of this groove is to retain a slide-in plexi piece and hold it in place - I intend to place some cushioning in the groove to prevent rattling. The most important feature for me is straightness - if it's not straight I will not be able to hold the plexi firmly and aesthetically in place.
This is a home project and I do not have a lot of tools. Specifically, I have no power tools apart from a drill/. I can't afford to buy a table saw (other than having a place to keep it) but could possibly buy a cheap angle grinder if need be - but I don't do a lot of DIY so I don't want the tool to be wasted.
What is the best way of creating such a groove, without splurging on single-use equipment?