Logic / maths puzzle involving distance and speed.?

Question

Ok try this for logic/maths. I did once ask my high school maths teacher but that was so long ago that I have no idea what he said, probably something like "tuck your shirt in boy and smarten up that tie".

A (super)man walks from point A to point B. Lets say 100miles.

Each step he takes halves the remaining distance.

However, for each step he takes he also doubles his speed.

For his first step he travels 50miles at 2mph.

When does he cross the line?

Sounds very impressive David. I take it that as this is an infinite series it allows for the fact that he never crosses the line, as half of something is always something.

Answer 1

Note that the first step, 50 mi at 2 mph, will take 50/2 = 25 hours. The second step, 25 mi at 4 mph, will take 25/4 = 6.25 hours. The third step, 12.5 mi at 8 mph, will take 12.5/8 = 1.5625 hours. Each step takes 0.25 times the duration of the previous step. This is an infinite geometric series where terms are of the form 25*0.25^k, so you just need to sum the duration of all the steps using the formula S = a / (1 - r), where a is the value of the first term, 25 hours, and r is the value of the common ratio, 0.25. a / (1 - r) = 25 / (1 - 0.25) = 25/0.75 = 33.333... hours.

As for the idea that Superman would never reach point B because his distance from the finish only ever decreases by half, leaving the second half yet to be crossed, that reasoning is invalidated by the fact that we have assigned Superman a finite speed that only ever increases. If he was traveling at 2mph to begin with, and never sped up, he would travel 100 miles in 50 hours. Therefore, at some arbitrarily later time, say 60 hours, he will have traveled 120 miles. If he always speeds up, as stated, then he will have traveled more than 120 miles. And if point B was only 100 miles away, he has clearly passed it. Since he has definitely reached and surpassed point B, we know that it is meaningful to determine when it happened.

Answer 2

Somehow I knew you were going to say that.

You reasoning (that he would never cross the finish line) is an offshoot of Zeno's Achilles and the tortoise paradox, which was proven false by Aristole some time back.

Go here:

http://en.wikipedia.org/wiki/Zeno's_paradoxes#Achilles_and_the_tortoise

To see the full paradox and the solution. . .

Answer 3

you're correct in answer a million. for 2), the documents we've is: entire distance = 1500m concern-loose entire velocity= 6 m/s (case a)) time elapsed = one hundred seventy seg to have an concern-loose velocity of 6m/s, he would desire to end the 1500m (a persevering with) in 250 segs ( t = d/v= 1500/6) he has already spent one hundred seventy segs, so he has 80 seconds left to run 500 m, the cost he would desire to attain is: v = d/t = 500/ 80 = 6.25 m/s ...its a similar for each and each of the scenaries they're asking you for