One of my favorite physics problems involves a turtle on a very wet rock. It may be that I really like turtles (which is true), but also it involves many of the best lessons in basic physics while still being somewhat intuitive.

The problem goes like this: Imagine a turtle sunbathing on a very wet rock. The turtle is balanced perfectly on the rock when a dragonfly buzzes by and very gently knocks the turtle to one side. Since the turtle is very slippery, and so is the rock, the turtle immediately starts sliding off the rock and into the pond.

Since we’re talking physics, we’ll go ahead and make some insane assumptions to allow us to solve this simply:

· No air resistance anywhere

· No friction between the rock and the turtle

· The rock is a perfect sphere

· The turtle is just a point

If we tried to just write down the laws of motion and follow the turtle down the side, this becomes a very difficult problem in spite of how intuitive the actual event may seem. The turtle starts sliding and at some point – not right away but before it hits the pond – the turtle is flying sideways more than the rock is extending sideways. At this point the turtle is airborne and remains so in a smooth arc until hitting water.

The insight that makes this problem a reasonable job is that the turtle will leave as soon as the net force points away from the center of the rock. We then just add up the forces and figure out when the force pointing into the rock’s surface is negative.

The force pointing into the rock is just the component of the gravitational force that points that direction. As the line connecting the turtle to the center of the rock becomes more horizontal, the effect of gravity along that line becomes less and less.

While this is happening, the turtle is gaining speed and the ‘centrifugal’ force – the apparent force pushing out on a spinning thing (i.e. turtle) is picking up.

When that centrifugal force – which comes from its speed along the surface – matches the radial part gravitational force – which keeps getting smaller as the line connecting the turtle to the center becomes more horizontal – the turtle flies off.

The last logical step here is figuring out how to relate those two. It turns out that all of the energy the turtle gains in speed it loses in height. Since the height is dependent on the angle, and so is the effective gravitational force along the radius, we’ve got a direct comparison. The rest is algebra.