A predator usually has greater mass and hence greater effect of inertia. It is more difficult for the predator to change direction compare to the prey.
We can spin an umbrella to remove water on the umbrella.
When the direction (of the fur/ umbrella) change, the water droplets keep on moving forward due to the effect of inertia, and hence leave the fur of the animals/umbrella.
Ways to Reduce the Negative Effect of Inertia
Seat belt and Airbag
The passengers in a car will be thrown forward when the car stop suddenly due to the effect of inertia. The seat belt and airbag can reduce the impulsive force acted on the body of the passenger and hence prevent serious injury.