2 Trains and Fly Problem. Find the number of trips made by the fly back and forth.

Question

Question: A Train A is approaching at a speed of 10m/sec, another Train B moving in the opposite direction at a speed of 20m/sec. A fly whose absolute speed is 50m/sec goes repeatedly from A to B and back, without loosing any time at any of the trains. The repeated moving back and forth stops when the 2 trains crash into each other. The initial distance between the 2 trains is 300m.

1>Find the distance traveled by the fly

2>Find the number of trips made by the fly back and forth

My Question: I get that the answer to the first question is 500m, since it takes 10 seconds for the 2 trains to crash, so in 10 seconds at a speed of 50m/sec the fly covers 50m/sec*10sec = 500m.

But how do I find the number of trips(back and forth) the fly makes until the trains crash?

Answer 1

Your answer to the first part is the intended one. It commits you to claiming the fly has no length and there are an infinite number of trips. You could compute the length of each trip. You would find a geometric series that when summed to infinity gives 500m. If there isn't an infinite number of trips, perhaps because you say the fly is crushed when the trains are closer than the length of a fly, you don't get 500m any more.

Answer 2

But an engineering solution might be to say that the fly stops going back and forth when the distance between the trains is equal to the fly's length, say $1$ cm. After all the work he's done, the poor guy starts getting smashed after this time. The distance between the trains could be expressed as, $D(t)=300-30t$, where $D$ and $t$ are in units of meters and seconds, respectively. To avoid the infinite result, we can calculate the trips the fly makes up until he starts getting smashed, $T_{smash}=\frac{300-\frac{1}{100}}{30}=9.999667$ seconds. Seems like the number of round trips the fly makes ($N$) can be approximated as:

$$ N=\frac{1}{2}\int_{0}^{T_{smash}} \frac{dt}{\frac{D(t)}{v}} $$ where $v=50$ m/s is the fly's speed and the $1/2$ multiplier is to make $N$ equate to 'round trips'. This computes to $$N\dot = 8.59.$$

Cheers,

Paul Safier