Being that it's a compatible E Mount, it will fit and work with the A5000, but it is a full frame sensor type lens as opposed to a regular E Mount asp-c crop sensor (smaller), that the 5000 and 6000 series cameras.
Tech specs aside, I don't know about using a fixed focal length lens for shooting video, but if you're shooting a baby that doesn't move around too quick like a toddler you should be able to get some good shots with beautiful depth due to the F/1.8, but I know from experience that chasing toddlers is easier with a zoom type lens.
On the budget concious side, using the 16-50mm pancake kit type lense would be a better choice.
If budget would allow (a few hundred dollars more), stepping up to the 18-105mm F/4 would be a great choice. Not nearly as compact as either the 50mm F/1.8 or the 16-50mm, but it's an awesome all around lens with a broad focal range that people have also said it's good for video use. I bought one to use with my A6000 and loved it, and use on occasion on my recently purchased A7Rii.