Put your hands up everyone who thinks that re-entering spacecraft heat up due to atmospheric frictional forces. Actually most modern spacecraft are blunt objects, and the air envelope that surrounds them upon re-entry generates little to no frictional force as there is not much relative velocity between the air and the spacecraft as it’s moving too fast for the air envelope to get out of the way. The spacecraft thus compresses the air envelope in front of it, and it is this compression of the air that causes it to heat up and ionise. This heat is then transported through the plasma envelope to the spacecraft’s surface by convection. So why does the opened parachute behind it travelling at initially the same speed not burn up either?

Throughput

The first problem was: “Prometheus stole fire from the gods which made Zeus very angry and he demanded that Prometheus return it to the gods 1:00. Then, to punish Prometheus’s further, Zeus issued the following set of orders to his infinite demons: “Demon #1, if Prometheus is alive at 2:00, kill him. Demon #2, if Prometheus is alive at 1:30, kill him. Demon #3, if Prometheus is alive at 1:15, kill him . . .” and so on, telling each subsequent demon to kill at half the time previously mandated.

But Prometheus is dead by 2:00, and the deities are not happy. They say: “Zeus, we’ll let you off the hook if you tell us which of your demons did this.” Zeus replies: “But none of my demons are guilty. Consider any one of them, and realize that he certainly could not have done it, for an infinite number of demons preceded him. Therefore it was none of mine.” So here’s the problem: Prometheus is dead, and yet for him to be so requires that there must be some first demon to have killed him, of which there is not. That is, he is dead and by the hands of no killer.

Prima facile Zeus appears right as it is Bernadette’s Paradox, an extension of Zeno’s Paradox. But, mathematically it appears possible. The timings given to demons are in a decreasing geometric progression 60, 30, 15,........60/2^(n-1), 60/2^n,....... in terms of minutes away from 1:00 given to 1st, 2nd, 3rd, ....., nth, (n+1)th,....demons respectively. Suppose it takes d minutes (d is quite small) for Zeus to give a command, then the time taken for commands to the demons are 0, d, 2d,....,(n-1)d, nd, .....minutes away from 1:00, an increasing arithmetic progression. Zeus chooses d and n such that 60/2^n<(n-1)d<=60/2^n. The command to nth demon is given at a time which is the time from 1:00 given to him or is just before the time given to (n+1)th demon. Surely, Prometheus is dead by 2:00. -- Abhay Prakash, (abhayprakash@hotmail.com

The premise of the Zeus, Prometheus problem is similar and in both cases the presumption is that the sum of the infinite series involved is infinite. However, the sum of all infinite series with finite elements are not infinite. There are infinite series, called convergent, with finite elements having finite sums. The series in both Zeno’s Dichotomy Paradox and the Zeus, Prometheus paradox is 1/2+1/4+1/8+1/16+1/32+ . . . which may be symbolically written as Sigma (n = 1 to infinity) (1/2)^n. This infinite series is convergent and its sum is 1. Therefore, though theoretically there are infinite intervals of time between 1’o clock and 2’o clock, their sum is finite and is equal to 1 hr. Which means that irrespective of the infinite intervals of time mandated to assassinate Prometheus, the total time elapsed from 1’o clock till the time Zeus mandated for the first demon, is 1 hr. Since the sum of the intervals of time mandated by Zeus is not infinite and has a finite value equal to 1 hr, Prometheus will be killed by one of the infinite number of demon’s by 2’o clock. -- Balagopalan Nair. K, balagopalannair@gmail.com

But Google This Now

A long pole helps a tightrope walker control his or her balance for reasons even economists don’t know. So would an ultralight nanofibre pole so long that it goes all the way around the Earth and meets up at the ends make a tightroper never lose their balance?

**Mukul Sharma**

**Sharma is a scriptwriter and former editor of Science Today magazine.**

**(mukul.mindsport@gmail.com)**