2。缺少条件

unless think in this way:
at moment t.
relative move to string is .01-(t+int(ds)|t)/L

L is string length at t, which is 1+t

it's a double integral problem. but think in extreme condition.
if the worm is not a dot but a length, one side will remain at origin and the other side will reach at end line at 100 sec time.
so the answer can be from 100 sec to infinity depends what kind of motion worm move together with string.