A car starts distance 1 from a wall then drives away at constant speed $ c $. There is a length of elastic tied between the wall and the car. Remarkably this doesn't affect the motion of the car (in reality the tension would slow the car).
Meanwhile, an ant crawls from the wall onto the elastic and then towards the car. On solid ground, the speed of the ant is $ a $. Needless to say, ants are slower than cars $ a \ll c $
Will the ant ever reach the car? Assume the elastic stretches without breaking.
Answer
Yes, the ant will reach the car. Calculus tells us exactly when:
Let C(t) be the position of the car at time t. The car travels at constant speed:
$ C(t) = 1 + ct $
Let A(t) be the position of the ant at time t. The speed of the ant is its speed on ordinary ground, plus a proportion of the car's speed (according to how far along the elastic it is):
$ \frac{dA}{dt} = a + { A \over { 1 + ct} } c = a + uc $
Making the natural substition $ u = {A \over {1+ct} } $ for the ants position as a proportion of the elastic. Then by the product rule
$ \frac{du}{dt} = { a \over {1+ct } } $
Which has solution
$ u = { a \over c } \log(1 + ct) $
Recalling that u is what proportion the ant is along the elastic, the ant reaches the car when u = 1 at
$ t = { {e^{c / a} - 1 } \over c } $
Edit: fixed missing c in exponent
I'd love to see an intuitive argument that the ant will reach the car without appealing to calculus.
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