183_projects:problem1_spring2021_4 [Projects & Practices in Physics]

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====== Project 4: Part B: Escape from Korath======

<WRAP info>
==== Project 4b: Learning Issues ====

  * Constant velocity and constant force models
  * Springs and spring forces 
  * Atoms and materials
  * Young’s modulus

</WRAP>

{{183_projects:korath.jpg?600|}}

You are a member of a team of scientists who have been working on a dimensional gate. After years of trying to find a sufficient power source to power the gate you get sent a mysterious power source discovered in the tomb of a Norwegian viking lord. After turning on the gate using the mysterious power source, you and your team gets sucked into another universe, which you quickly figure out is the Marvel Comics Universe. 

You are arrested and set to be executed for being enemies of the Kree Empire. You quickly manage to escape, but you are being chased by Korath the Pursuer. Along with your trusty stopwatch, a thermometer, and an apple for sustenance, you also brought along a piece of tech that alerts you of another operational dimensional gate at the bottom of a snowy cliff. You reach the edge of the cliff, which is apparently named the "Cliff of Insanity." You look over the edge and through a dense fog. You notice a metal platform that is positioned at the bottom of the cliff but you can't judge this distance by eye. The platform is rigged with an electronic switch that will trigger if any metal comes into contact with it - activating an alarm that will alert Korath of your location.

Fortunately, the cliff top has a supply shed that contains three very large spools of unusual metal cables composed of adamantium, unobtainium, and vibranium; each cable has a diameter of 2.0 cm. In the shed is also an incredibly strong pair of metal shears, which can cut the cables to any length, and a clamping mechanism that can be used to attach one end of a cable to the edge of the cliff. To escape you need to climb down one of these cables to the platform. What material and length will do the job?

You pick up the metal cables and notice that each is quite stretchy; the table below gives the length of the bond between two adjacent atoms ($d$) and the stiffness ($k_{s,i}$) of a single interatomic spring. Each element has a simple cubic crystal structure.

^ Element ^ Bond Length (m) ^ Stiffness of Single Interatomic Spring (N/m) ^
| Adamantium | $2.3\times10^{-10}$           | $4.7\times10^{-2}$         |
| Unobtainium | $3.5\times10^{-10}$            | $3.8\times10^{-2}$          |
| Vibranium | $4.3\times10^{-10}$            | $1.2\times10^{-2}$          |