One of Crazytown’s nuclear bombs has just gone into a critical meltdown. There’s no way to defuse it so the only solution is to contain the blast. The mayor has hatched a crazy plan to stash the bomb in the biggest container ship the city has, fill it with concrete, and then let it sink to the bottom of the ocean as far offshore as they can make it.
She puts you in charge.
You need to calculate how much concrete is needed, and how far offshore you can make it in the 4 hours remaining before the bomb warhead detonates. Loaded up, the ship has a top speed of 4 km/hr, and will sink at a rate of 2 metres per second. The ocean you’re sinking the ship in is 5 kilometres deep.
The central storage area in the ship is a rectangular prism with the dimensions above. Concrete has a density of about 2.4 tonnes per cubic metre, so you’ll need to work out the weight of concrete you need to completely fill this volume up.
Volume = L x W x H
Volume = 300 x 40 x 30
Volume = 360,000 m^2
Once you have the volume, you can multiply it by the density of concrete to get the total weight of concrete you need:
Concrete required = volume x density
Concrete required = 360,000 x 2.4
Concrete required = 864,000 tonnes
That’s a lot of concrete. You tell the mayor what you need.
Next you need to work out the timing. There are two stages to the journey – the journey out to sea away from the city, and then the journey to the bottom of the ocean. You have a total of 4 hours for both.
Let’s work out the sinking time first:
Sinking time required = depth to sink / rate of sinking
Sinking time required = 5000 m / 2 m/s
Sinking time required = 2500 seconds
That leaves you a little over 3 hours to motor the ship as far away from the city as possible first. The exact time is:
Time to escape the city = 4 hours – 2500 seconds
Time to escape the city = 4 x 3600 seconds – 2500 seconds
Time to escape the city = 11,900 seconds
Time to escape the city = 3.306 hours
Let’s work out how far you the ship can travel in that time at its top speed:
Ship travel distance = ship speed x time
Ship travel distance = 4 km/hr x 3.306 hours
Ship travel distance = 13.22 km
Real-life Example – Nuclear Testing Site Bikini Atoll
Bikini Atoll is a tiny atoll (a coral reef and lagoon) in the Marshall Islands that was used for a large number of nuclear tests from 1946 onwards.
One of the most powerful devices tested there was called “Castle Bravo”. It was particularly destructive because the scientists made a theoretical error when calculating how powerful they thought the explosion would be. As a result, it was three times as powerful as they had predicted, which led to unexpected radioactive fallout and contamination.
The test was equivalent to exploding about 15 MT – 15 million tonnes – of TNT, and created a fireball about 7 km in diameter. If that had occurred in this scenario, it would have vaporized the water in a large ball more than halfway to the surface of the ocean 5 km above.