This type of case does come up in the news once and awhile. When the person is found that shot the goose/duck( or other animal), and the arrow head used is looked at, there is two things is common. The shooter was NOT a hunter, and the arrow point was a target point not the right type to use for hunting.
To the OP I will give some advice. Try to hit the goose in the top/front 1/3 of the body just below the wing, and behind the shoulder. If you look at a picture of a standing goose and draw a line from the base of it's neck to it's leg you will see a spot just under the wing and behind the shoulder. This spot is where the heart and lungs will be. Use broad heads with two blades, or smaller three blade ones back by a collar to add blunt force to the impact.
DO NOT use a blunt, a field point with a collar, or the G5 SGH. They will not penetrate enough of cause enough bleeding for a quick kill.
These are the collar that I have use with success.
http://cdn-us-ec.yottaa.net/5208e105...ocs=_&yoloc=us
http://www.ewbateman.com/images/Misc...s/ADDLarge.jpg
So your arrow should look like this...
http://www.3riversarchery.com/images/medium/4205X.jpg