A prey animal is in the middle of a circular pond. Outside the pond is a predator. The predator can not swim.

The movement speed of the predator is 4 times faster on land then the movement speel of the prey animal in the water.
The pray animal is faster on land.

Can the pray animal escape the predator?

On a simplistic level, if the prey animal keeps swimming in a straight line away from the predator’s starting position, it won’t reach the shore in time because the predator will have run pi times r, and pi = 3.14159…

But if the prey animal continuously changes course to swim in the opposite direction to the predator’s new position, then it may be possible for it to escape. But it’s a long time since I did maths in school and I’ve no idea what the formula would be for the distance travelled by both prey and predator in that scenario. It’s probably a calculus problem, but then again, I could be totally wrong.

Assuming the pond has radius 1, the prey animal can move to the circle of radius 1/4 and maintain a position opposite the predator.
This is possible because the prey only has 1/4 the distance to travel.
When the prey then makes a mad dash to the edge of the pond farthest from the predator, the prey has to travel a distance of 3/4.
The time for the prey is 3/4 and for the predator is pi/4. Since pi/4 is greater than 3/4, the predator escapes.

If the prey were stupid and ran in a straight line from the centre of the pond, he would get caught.