Obviously, if you own a cheetah, your cat can outrun a grizzly, but that's cheating. This is about domesticated kitties, the kind that more easily fit in a bear's flesh-rending maw. So how fast would one of those have to be? The Best of Yellowstone National Park says a grizzly bear can top 40 miles per hour. Is any house cat that fast? No. So that's a sad ending.
Per the Spruce Pets, the fastest domestic cat on the planet is the Egyptian Mau. It's spotted kind of like a cheetah but it only runs 30 miles per hour. That's nothing to sneeze at but compared to a grizzly, this isn't even a case of "close but no cigar." The cat would just get smoked, and bears probably find smoked meat delicious. Anyway, the Mau is said to be loyal, friendly, and great with children, so maybe you would fight that big old grizzly to protect your precious Egyptian Mau.
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