An injection through her heavy clothing? Dropping the needle for someone to pick up? Doubtful.
Nevertheless it's true - she was frozen and couldn't walk when the the car pulled up. So: she needed the injection immediately to unfreeze her from the seizure. Once injected however, she collapsed and had to be held up and put into the car, so the injector was dropped to do that.
Watch the video - it happened in mere seconds.