Tunnel Vision Variables

E4
Think of a time 300 years from now, when, thanks to global warming, the Arctic and Antarctic ice sheets have melted and there's only ocean on top and land down under. In order to replace fossil-fuel burning transport, people have dug a hole 20m in diameter all the way from what used to be the North Pole to the South Pole and constructed an enormous gravity transport tube. Through this tunnel, cylindrical cars carrying both supplies and people are dropped from one end to the other. All friction has been eliminated by maintaining a vacuum inside the tube, and by using magnetic fields, the cars are kept away from the tube's sides. The pole-pole trip now takes only a little over 42 minutes. How many of the following seven questions - in increasing order of difficulty - can you answer?
(1)As the car travels from the North Pole to the Earth's centre, does its velocity increase, decrease or stay the same? (2) At the same time, does its acceleration increase, decrease or stay the same? (3) If you happen to be riding in one such car and it stops when it's exactly half way down to the Earth's centre, would you weigh less or more on a spring scale than on the Earth's surface? (4) At what point during the trip would you experience zero gravity? (5) At which point does the car reach maximum speed, and how fast would it be going at that time? (6) If cigarettes weren't banned by then, would smoking be allowed? (7) If a car fell down a similar tube through the centre of the Moon, would the time for a one-way trip be longer or shorter than 42 minutes? Also, what about Mars?
 DEAR MS
(The problem was: "A bag contains a counter, which may be either black or white. A second counter, which is definitely white, is put into the bag. The bag is shaken and one counter taken out, which proves to be white. What is the probability of the next counter coming out of the bag also being white?" - MS)
Black-&-White Dept:
The simplest way to go about it is to magnify the problem 100 times. Now, in the first draw, we'll have 75 whites and 25 blacks. Ignore the 25 blacks as they do not qualify for the second draw. In the second draw, we have 50 whites and 25 blacks left. Ergo, white has a 66.6 per cent chance.

• Ravinder Nair, ravin866@gmail.com