Can someone help me with this math problem and show me how to answer it?

Question

Suppose you have 2 dogs and 1 flea. The dogs spy one another from opposite ends of a 100 ft driveway & begin to run towards each other, each at a speed of 10 feet per second. Starting on one dog, the flea springs back and forth from nose to nose at a speed of 20 feet per second until the dogs collide and flatten him. How far did the flea travel in all?

Answer 1

The dogs travel for 5 seconds ( 100' / (10'/s + 10'/s) ) so the flea travels 100 ft in 5s (5s * 20'/s)

Answer 2

Two ways to solve this one. One is a complicated method involving summing the infinite converging series of flea trips.

The other method is easier. The time it takes the two dogs to reach each other is 5 seconds (each is going 10 fps, so they close at 20 fps).

During the 5 seconds the flea is traveling at 20fps, so it travels a total of 100 feet.


Just in case you are interested in the complicated method, the flea travels twice as fast as the oncoming dog, so on each trip it travels 2/3 of the remaining distance. First trip is 66.666.. ft. 1/3 of the remaining distance is left between the dogs (each has travelled 33.333... ft after the first trip).

So, the infinite sum is:

100 (2/3) + 100 ( 2/3 ) ( 1/3 ) + 100 ( 2/3 ) ( 1/3 )^2 + 100 ( 2/3 ) ( 1/3 )^3 + . . .

This is (200/3)(1 + 1/3 + 1/9 + 1/27 + ...)

For the infinite series factor, you can just solve for x in: 3(x-1) = x, which is x = 3/2. And then:

(3/2) * (200/3) = 100 ft

Answer 3

Speed is relative. If the flea is traveling at 20 feet per second we must first ask, "20 feet per second relative to what? Relative to one dog? The other dog? The ground?"

If we assume it is relative to the ground, then these simpletons will be almost right, discounting relativity (5seconds * 20 feet/second = 100 feet).

If we assume the flea's speed is relative to the nose from whence it jumps, then it is traveling at 30 feet per second relative to the ground and will travel 5*30 = 150 feet, again discounting the theory of relativity which will modify the answer depending on the observer's speed relative to the flea circus.

Answer 4

Each dog only has to travel half way, so 100/2=50 feet. 50 feet / 10 ft per sec = 5 sec. So far each dog is travelling 5 seconds total. 20 feet per second times 5 seconds would be 100 feet travelled.