Two Trains and A Fly Puzzle

Classical Mechanics Level 4

Two trains are on the same track a distance $100\ \text{km}$ apart heading towards one another, each at a speed of $50\ \text{km/h}$ . A fly starting out at the front of one train, flies towards the other at a speed of $75\ \text{km/h}$ . Upon reaching the other train, the fly turns around and continues towards the first train. It continues flying back and forth till the two trains meet and it gets squashed. The time taken (in hours) for the fly to make $7$ 'rounds' is $\color{#3D99F6}{\dfrac{p}{q}}$ , where $\color{#3D99F6}{p}$ and $\color{#3D99F6}{q}$ are positive coprime integers. Calculate $\large\color{#3D99F6}{p+q}$ .

Note : A round is defined as moving from one train to another.

The answer is 156249.

1 solution

Saket Joshi
Oct 14, 2014

Solve it in the frame of the fly. In the frame of fly: One train is always approaching with 125 and one is moving away with 25.

figure out t1 i.e. distance for 1st collision (100/125 =4/5);

distance from fly travelled by the other train in t1= 25*t1;

time to travel back to other train t2= (25*t1)/125 =t1/5;

similarly time between r th collision is t1/r

So required = t1 + t2 ... t7

                  = t1 [ 1 + (1/5) + ... (1/5)^7)]

                  = (4/5) [ ( (1/5)^7 -1 ) / ( (1/5) -1) ]

Solve to get 5^7 -1 / 5^7 that is 78124/71825

Hence answer is 156249

Now, where is my prize?

Just the same method using calculator for 5^7 etc..

Niranjan Khanderia - 4 years, 10 months ago

Two Trains and A Fly Puzzle

The answer is 156249.

1 solution

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