Tuesday 19th May 2:00 pm
How long would it take to fall through the Earth?
Tuesday 19th May 2015 2:00 pm
If you could jump through a hole in the Earth, how long would it take to get to the other side? Dr Karl digs up the maths.
Suppose you drilled a hole right through the centre of the Earth and out to the other side — and then you jumped into the hole. What would happen?
The answer used to be that you would pop out on the other side of the Earth 42 minutes later. But a bit of fact-finding and fine-tuning tells us that it’s closer to 38 minutes.
Yup, 38 minutes to fly from one side of the Earth, right through the middle, and out to the other side.
First, let’s assume you could actually build this tunnel. You would need both the drilling machines, and the walls of the tunnel, to be made of some incredibly exotic material such as ‘unobtainium’.
As you drill down towards the centre of the Earth, you would first pass through up to 100 kilometres of solid crust, followed by a few thousand kilometres of molten rock.
At 2900 km down you would plunge into some 2200 kilometres of liquid iron and then smack into about 1300 kilometres of solid iron. Down at the core, about 6400 kilometres down, the temperature would be hotter than the surface of the sun. Only unobtanium could withstand this truly hostile environment.
Second, let us make believe there is no air in the tunnel. Obviously, if there was air, the wind resistance would slow you down. But more importantly, once you’ve fallen several hundred kilometres down the vertical tunnel, the weight of the air above you would have generated so much pressure that the air around you would have turned from a gas to liquid and then a solid. You couldn’t fall through that.
Third, let’s blissfully ignore the fact that the Earth is rotating. On the surface of the Earth at the equator, you are moving from east to west at about 1700 kilometres per hour. By the time you had reached the centre of the Earth, you would be grinding against the side of the tunnel with that same sideways velocity of about 1700 kilometres per hour.
Fourth, let’s assume that the Earth has a constant and average density of about five-and-a-half tonnes per cubic metre.
So now that (in our imaginary perfect world) we have our airless frictionless tunnel, let’s jump in. The entire mass of the Earth is under us, pulling on us with its gravity. By the time we reach the dead centre of the Earth, we are moving at about 28,000 kilometres per hour.
If we could magically come to an instantaneous dead halt, we would just float, free of gravity. The gravitational pull of the Earth around us would be effectively zero.
Directly ahead of us would be the mass of half the Earth — but that would be exactly cancelled out by the mass of the other half of the Earth that was behind us.
But once we’ve reached the core, we have a tremendous velocity, and so we keep moving. However, as we journey upwards, there is an ever-increasing amount of our planet behind us as we move further from the core, and its ever-increasing gravity slows us down. But the amount of energy we picked up on the way in, is exactly what we need to propel us upward against the pull of gravity.
After a total time of 42 minutes and 11 seconds we appear on the other side of the Earth. If we grab onto the side of the tunnel, we would be able to haul ourselves out. But if we don’t, we’ll return to our starting point another 42 minutes 11 seconds later. Like a pendulum, we would oscillate back and forth indefinitely, for ever and ever and ever …
But in 2015, Alexander Klotz, a student from McGill University in Canada, realised that the density of the Earth is not constant — it’s less at the surface, and greatest at the core. The density of our planet is about 1 tonne per cubic metre at the surface, but rockets to 13 tonnes per cubic metre at the core. Furthermore, about halfway down to the core, there’s a rather dramatic jump in density near the boundary where the ball of liquid iron begins.
So when Mr Klotz carried out the calculations with the new data (sure, you can look it up), it turned out that the time to fall through a hole in the Earth is about 38 minutes and 11 seconds – four minutes quicker.
Now Mr. Klotz begins his third paragraph with the words, ” … it is unlikely that such a tunnel will be excavated in the near future …”. Sure, it’s not ‘shovel-ready’, but near future? Does he know something the rest of us don’t?
Only the passage of time will tell …
This blog first appeared on Dr Karl's Great Moments in Science
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