Boarder collies are very VOCAL dogs. I attended flyball training recently and the collies were barking constantly - in the vans, having down time and resting, running the course, all for the whole two hours. This is an environment they train in every single week - TRAINING is NOT the issue.
Give your dog an alternative, mutually exclusive behaviour to carry out in this situation. This will require your mother's input, a few treats or other reinforcer and a bit of time each day for a few weeks. You could teach anything as an alternative behaviour, such as going to "place" in another room, fetching a ball for you to throw, anything else you can think of that will prevent the barking. First, teach the behaviour with a verbal cue and hand signal. Then, slowly transition to cues into your mum standing up from a chair. It might take a few tries for your dog to finally "click" and get it. If they get completely lost, go back a few steps for a few days. This will be a long-term fix, so is totally worth the little time and effort in comparison.
When they CONSISTENTLY preforming the behaviour when your mum gets up from the chair and you have lowered the value of the reward (so they aren't dependent on reward each time, also keeping it interesting by changing the reward) you will need to "proof" the behaviour. Do this by having your mother change chairs, change rooms and add distance. You hide out of sight as well, so the behaviour isn't dependent on you being there. Be sure to use a clicker or marker word, followed by at least verbal praise EVERY TIME your dog is successful.
For a lot of these alternative behaviours, you may need to break it down into little teaching steps. Taking going to place for example, teach the dog to go to place first. At any distance, changing the position of place (a blanket or bed), adding suitable distraction such as another person or dropped food on the floor (only ask your dog to do things you think they will succeed at. Increasing how successful your dog is will allow them to have a better understanding of what is being asked and they will stay motivated as they are frequently receiving rewards). Once place is learnt, teach your dog the hand signal to go with it. Then use just the hand signal (to be sure they have learnt it, not just following the verbal cue). You could have your mother then use the hand signal and verbal cue as she stands up, then fade both out, until just her standing up from the chair works. You might use other combinations - whatever works for your dog. Go at their pace for better success.
Good luck, persevere and build a better relationship with your dog! It may help if your mother understands that barking is not always aggressive - my dogs bark as an invitation to play (accompanied with a play-bow usually). My older dog started barking non-stop at my mother earlier as well. He had spotted a cat in the garden and was trying to indicate this to us (I taught him to bark at cats in the garden, to keep the rabbits safe when they're free-ranging. He loves cats in all other circumstances :D). But of course, he had never had to do this from inside before and as dogs generalize badly, he didn't know that barking at the window or at me (who taught him) would've been easier for us to understand!