## Question:

A spider has fastened one end of a ‘super-elastic’ silk thread of length 1 m to a vertical wall. A small caterpillar is sitting somewhere on the thread. The hungry spider, whilst not moving from its original position, starts pulling in the other end of the thread with uniform speed ${{v}_{0}}=1cm{{s}^{-1}}$. Meanwhile, the caterpillar starts fleeing towards the wall with a uniform speed of 1 mm ${{s}^{-1}}$ with respect to the moving thread. Will the caterpillar reach the wall?

## Step-by-step

The velocity of the thread at a distance of x meters from the wall is obviously proportionately smaller than the velocity of the end of the thread,i.e. it is $x{{v}_{0}}$.
If this value is greater than the speed of the caterpillar, then the latter will move away from the wall. Its situation will become more and more hopeless, and it will never reach the wall.
On the other hand, if ${v}_{caterpillar}{>x}{{v}_{0}}$, the net velocity of the caterpillar is towards the wall and increases as time passes, with the consequence that the caterpillar will certainly reach the wall. The limiting case corresponds to $x={v}_{caterpillar}/{{v}_{0}}=0.1m$. Starting at this point, the caterpillar does not move in either direction.