You actually need to be pretty careful in these situations. My father is now a convicted felon because he shot a cat.
He rented out a house on his property in rural WA state, the tenants put out food to attract wildlife, and a feral tomcat showed up. My dad had just purchased two kittens for mouse control, and they lived in his garage. The feral tomcat came next door and claimed the garage, killed one of the kittens and tore an ear off the other. My dad went to talk to the neighbors and they claimed ownership of the cat, so he gave them written notice to keep it on their property, and being my dad of course he told them "If it's in my garage again I will shoot it."
A few nights later of course it showed up to finish off the other kitten, so he shot it with a 12ga, but since it was running it wasn't a clean shot. It ran into the brush, the neighbors called the cops, cops showed up, arrested my dad for animal cruelty. They said that he should have made every effort to find it and put it out of it's misery quickly, it did eventually die from the wound but took a long time and was crying from somewhere in the brush while the cops were on scene. It's not very feasable to chase a cat through thick WA State brush on property as steep as my dad's place, especially in the dark. The cop also admitted in court that he would not have been able to find the exact location of the cat, but could only hear that it was in pain.
Long story short, 60 days in jail, felony conviction, and all the BS that goes with that. We live in a sick world, the tenants had no vet paperwork or anything but were able to claim that it was their "pet" that they cared deeply about and were horribly traumatized by the experience. Nevermind that it was shot IN my father's garage, caught in the act of trying to kill an actual pet, and after written notice that it needed to be controlled was given.
Be careful what you do, it's not worth ending up in jail for a stray cat.