Henry Ernest Dudeney/Puzzles and Curious Problems/72 - The Fly and the Motor-Cars/Solution

by : $72$

 * The Fly and the Motor-Cars

Solution

 * $(1): \quad$ At $13:48$, at which point they are $120$ miles from $A$.


 * $(2): \quad$ At $14:00$, at which point they are $100$ miles from $A$.

Proof
Let $d_1$ miles from $A$ be the point at which the fly first meets $A$.

Let $d_2$ miles from $A$ be the point where $B$ is at that time.

Let $t_1$ hours after noon be the time at which this happens.

Let $d_3$ miles from $A$ be the point at which the fly then meets $B$.

Let $t_2$ hours after noon be the time at which this happens.

First we investigate where everybody is when the fly first meets $A$.

We have:

So when the fly first meets $A$, they are $75$ miles from $A$, while $B$ is $150$ miles from $A$ (which also happens to be $150$ miles from $B$).

This happens at $1 \tfrac 1 2$ hours after noon

Now we investigate when the fly first meets $B$.

At this stage we are not interested in what happens to $A$, just $B$ and the fly.

We have:

So the fly meets $B$ at $120$ miles from $A$, which happens at $1 \tfrac 4 5$ hours after noon, or at $13:48$.

Let $d_4$ miles from $A$ be the point at which the cars collide.

Let this happen at $t_3$ hours after noon.

We have:

Thus $A$ and $B$ collide at $2$ hours past noon, $100$ miles from $A$.

We do not need to calculate the path of the fly, despite the fact that the naïve reader may be tempted into trying.

Also see

 * The Bulldozers and the Bee, of which this is a variant.