# Two Trains (Constant Velocity) and a Flying Creature

Two trains are headed toward one another on the same track. Train A has a constant speed of 40[km/hour] and Train B has a constant speed of 20[km/hour]. A Tengu that can fly at 60[km/hour] is perched on top of Train A. When the trains are 90[km] apart, it hops off and begins to fly toward Train B. When it reaches Train B, it instantaneously reverses direction and begins flying toward Train A. How far does the Tengu fly before the trains collide?

Do not make this harder than it is. A series of increasingly small distances would fit this problem perfectly, but that is needlessly complex.

The trick here is to understand that the “Flying Creature” is always moving at a constant velocity and merely bounces off of an imaginary plane in front of the oncoming train before heading back to the other train. Because of this assumption, the Tengu will cover the same amount of ground in the time before the trains collide flying between them as if it had been flying in a straight line for the same amount of time.

First, calculate the time between the start and when the trains collide. The idea is that at some time in the future, the distance traveled by Train A and Train B added together will equal the original distance between them at the start.

$t$ = Time Before the Trains Collide [hour].
$V_{A}$ = Velocity of Train A [km/hour].
$V_{B}$ = Velocity of Train B [km/hour].
$D$ = Distance Between the Trains [km].

$(V_{A}*t)+(V_{B}*t)=D$
$(40[km/hour]*t)+(20[km/hour]*t)=90[km]$
$t=1.5[hour]$

Now with time in hand, it is simple to calculate how far the Tengu flew in the alloted time.

$V_{T}$ = Velocity of the Tengu [km/hour].
$H$ = Distance Traveled by the Tengu [km].
$V_{T}*t=H$
$60[km/hour]*1.5[hour]=90[km]$

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