E/C MAC1105 Spring 2018 Tsunami Project

E/C MAC1105 Spring 2018 Tsunami Project
(8 points)
NAME __________________________________
Tsunamis are destructive natural events which fortunately do not occur that often, but when they do, they leave much damage behind. In this project you will learn a bit of the mathematics used to determine their impact. You will want to visit the following website:
https://www.tsunami.gov/?page=tsunamiFAQ
In section 3.2 of the webpage above, a mathematical model of the wave’s speed is provided and states that a tsunami wave traveling over waters that are ℎ feet deep is approximately 𝑣=√𝑔ℎ where 𝑔 is the acceleration due to the force of gravity, whose value is 32 𝑓𝑡/𝑠2 [feet per second per second].
Assume that an underwater earthquake 800 miles off the coast of Iwaki, Japan, produces a tsunami (which spreads in all directions and therefore it is headed for Iwaki!) see figure below:
The depth markers are placed in equally spaced intervals and indicate the ocean depth (in feet) at that location (for example, the 10,000 marker indicates the ocean depth at the location of the earthquake).
(a) Calculate the tsunami’s speed at each of the markers (complete the table below). The speed you obtain applying the model will be in feet/second. You will need to convert that to miles per hour, which are units more familiar to us. The conversion factors are: 1 𝑚𝑖𝑙𝑒=5280 𝑓𝑒𝑒𝑡 and 1 ℎ𝑜𝑢𝑟=3600 𝑠𝑒𝑐𝑜𝑛𝑑𝑠. Thus, to convert 400 𝑓𝑡/𝑠𝑒𝑐 to miles per hour, we proceed as follows:
400𝑓𝑡𝑠𝑒𝑐×𝑚𝑖5280 𝑓𝑡×3600 𝑠𝑒𝑐ℎ𝑟≈𝟐𝟕𝟐.𝟕𝟑𝑚𝑖ℎ𝑟
Use this procedure to compute the speeds and enter your numbers in the table (please show all your calculations).
Depth marker
Wave speed (mi/hr)
10,000
7,500
5000
3120
10
10000 ft 5000 ft 7500 ft
Iwaki 3120 ft 10 ft
800 𝑚𝑖𝑙𝑒𝑠
(b) The impact a moving object has on whatever it collides with depends on both its speed and its mass. For example, if a ball of mass 0.5 kilogram hits you at a speed of 4 meters per second (like a soccer ball), the impact is negligible (unless you get hit in the face!). But, if a four-ton (4000 𝑘𝑔) truck hits you at the same speed … you fill in the details!
The quantity that measures the combined effect of speed and mass is called momentum. Momentum is defined by the equation 𝒑=𝒎𝒗 where 𝑚=𝑚𝑎𝑠𝑠 and 𝑣=𝑠𝑝𝑒𝑒𝑑. Technically, 𝒗 is “velocity”, and you will learn about that if you take a basic physics course – pre-med and science majors will have to take such a course.
Let us model the tsunami as a wall of water 50 meters long by 1.5 meters high by 10 meters wide as it approaches the shores of Iwaki. The density of pure water is 1000 𝑘𝑔/𝑚3 and that of ocean water is even greater due to its salt content, but let us take 1000 𝑘𝑔/𝑚3 as an approximation.
Calculate the momentum of the wall of water just before it hits the beach (at the 10 ft. marker). Remember: 𝑚𝑎𝑠𝑠=𝑑𝑒𝑛𝑠𝑖𝑡𝑦×𝑣𝑜𝑙𝑢𝑚𝑒. The volume of water in the wave is the product of the dimensions given above, and to express momentum in metric units (S.I. units), we will need to convert the speed in 𝑓𝑡/𝑠𝑒𝑐 to 𝑚𝑒𝑡𝑒𝑟/𝑠𝑒𝑐𝑜𝑛𝑑. We do this with the conversion factor 3.281 𝑓𝑒𝑒𝑡≈1 𝑚𝑒𝑡𝑒𝑟. Thus, a speed of 60 𝑓𝑒𝑒𝑡/𝑠𝑒𝑐𝑜𝑛𝑑 is converted to a speed in 𝑚𝑒𝑡𝑒𝑟𝑠/𝑠𝑒𝑐𝑜𝑛𝑑 like this: 60𝑓𝑡𝑠𝑒𝑐𝑜𝑛𝑑×𝑚3.281 𝑓𝑡≈18.29 𝑚/𝑠𝑒𝑐
Now calculate the mass of the water wave and convert its speed at the 10 𝑓𝑡 marker to a speed in 𝑚/𝑠𝑒𝑐. The product of these to quantities will give the wave’s momentum (units of 𝑘𝑔∙𝑚/𝑠) just before it hits land, and it will be a large number. For comparison purposes, the momentum of an 18-wheeler traveling at 70 miles per hour is approximately 977,571 𝑘𝑔∙𝑚/𝑠𝑒𝑐. How would you like to be in front of a tsunami wave? Please submit all your calculations.
As an aside, of course the number you got is large. But to be fair, this momentum is distributed along the face of the wave: 50 meters long by 1.5 meters high. The momentum per unit area therefore would be the value you found divided by 50×1.5=75 𝑚2. Calculate that value and assess the potential damage the wave would cause to life and property that stands in its way.