This worksheet is based on events in the mathematical thriller A Question of Will. Get
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www.MathNovel.com FireballRadiiWorkedSolution
Question1
If the fireball radius scales linearly with the explosive yield, we need to work out the yield ratio between the 100 tonne bomb and the 15000 (15 kilotonne) bomb: Yield ratio = 15000 / 100 Yield ratio = 150 (1 mark) We can then multiply the fireball radius for the 100 tonne bomb by this yield ratio: 15 kT fireball radius = yield ratio × 100 tonne fireball radius 15 kT fireball radius = 150 × 30 m 15 kT fireball radius = 4500 m = 4.5 km (1 mark) Question2
If the fireball radius scales with the cube root of the explosive yield, we need to find the cube root of the yield ratio we calculated in the last question and use this to scale up the fireball radius. We can estimate the cube root of 150 first: 5 × 5 × 5 = 125 – smaller than 150 6 × 6 × 6 = 216 – larger than 150 So an estimate of the cube root of the yield ratio is between 5 and 6 – let’s say 5.5. If we use this estimate to scale up the fireball radius, we get: 15 kT fireball radius estimate = (yield ratio)1/3 × 100 tonne fireball radius 15 kT fireball radius estimate = 5.5 × 30 m 15 kT fireball radius estimate = 165 m (1 mark) Now we can use a calculator to calculate the cube root precisely: 15 kT fireball radius = (yield ratio)1/3 × 100 tonne fireball radius 15 kT fireball radius = 1501/3 × 30 m 15 kT fireball radius = 159.4 m (1 mark) This answer is pretty close to our estimate. Question3
We need to repeat the calculation performed for the last question, but using the yield ratio raised to the power 0.4 instead: 15 kT fireball radius = (yield ratio)0.4 × 100 tonne fireball radius 15 kT fireball radius = 7.421 × 30 m 15 kT fireball radius = 222.6 m (1 mark) Notice how a slight change in the power – changing it from 0.3333 to 0.4, makes a fairly big difference in the calculated fireball radius. All content copyright Michael Milford. See www.MathNovel.com for more information. This worksheet is based on events in the mathematical thriller A Question of Will. Get
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First we calculate the new yield ratio: Yield ratio = 50000000 / 100 Yield ratio = 500000 (1 mark) Then we use it and the 0.4 power law to work out the fireball radius for the Tsar Bomba: 50 MT fireball radius = (yield ratio)0.4 × 100 tonne fireball radius 50 MT fireball radius = 500000.4 × 30 m 50 MT fireball radius = 5711 m (1 mark) We can see how much bigger our calculated radius is then the actual observed radius of approximately 4 km: Calculation to observed ratio = 5711 / 4000 Calculation to observed ratio = 1.428 So our calculated estimate is about 40% larger than the actual observed estimate. (1 mark) Finally, we need a map of a city – I’m using OpenStreetMaps for the city of Brisbane, Australia, and a scale on the map, so we can superimposed a circle with a radius of 4 km: (1 mark) All content copyright Michael Milford. See www.MathNovel.com for more information.