An Unhappy End

Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...

Age

11 to 14

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Two engines are 1000 metres apart at the ends of a single track railway line.

They set off towards one another at 10 metres/second.

Just at that moment, a fly sitting on the front of one of the engines sets off and starts to fly along the railway line at 25 metres/second.

The fly eventually meets the front buffer of the other engine which by now has travelled some way along the track.

The fly immediately turns round and flies at its usual speed towards the first engine.

When it gets there it turns round again and flies back towards the other engine, and so on, and so on...

How far will the fly have travelled before the unhappy end?

Why do this problem?

This problem only requires some simple understanding of the relationship between time, distance and speed, but it will require clear thinking and insight to solve. It provides a good experience of the mathematical 'lightbulb' moment when a seemingly tricky problem is solved using a clever trick or alternative viewpoint.

Possible approach

Show the 'film' asking pupils to focus on the fly. Ask them what data they would need to analyse the situation mathematically, and provide the speeds and distance.

Ask students to begin calcuating times, distances and positions of the trains and the fly. They will need to establish a clear way of recording their results. Encourage them to describe their strategy, even if they are frustrated by the length of the task.

It is useful to discuss with students that in some circumstances, the only option for solving a problem is to find a long, complex algorithm. However, the frustration CAN be a trigger to step back and find more efficient methods, and in this case there is a good method.

The simple way of looking at this problem is the total time until collision of the two trains and the complicated way is working out the points at which the fly hits a train on each part of the journey and then adding these up. Ideally someone in the group will come up with the simpler method - if not, you could leave it as an open problem, or begin to encourage the group to focus on time.

Key questions

How long will it be until the fly first hits a train?
How long will it be until the trains collide?

Possible extension

Alter the distance between the engines or the speeds to change the problem.

If the fly flew for 1 minute, what could the distance and speeds have been?

If the fly flew 2000 metres, what could the distance and speeds have been?

Speedy Sidney provides a similar context for a slightly more challenging problem.

This problem is a perfect opportunity to think about infinite series - the fly makes infinitely many trips between the two trains before collision. The total time taken for all of the trips must equal the time taken for the trains to collide. Students might wish to investigate the creation of this series by working out how long the fly takes to travel on each portion of its journey.

Possible support

As a warm up, you might try the simpler problem Bike Ride

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically