Disclaimer: We could have made this about scientists or engineers testing something in a lab and the many applications of springs or the spring model but there is a point at which lab testing gets a bit boring so we had a little fun with the story.

Forgive us.

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.

Learning Goals Week 4

Success! Your team has made it to the bottom of the “Cliff of Insanity” without alerting Korath the Pursuer to your location. Unfortunately, the piece of tech the team has, which guides you to the nearest dimensional gate, must have been malfunctioning at the top of the cliff because there is not a single-dimensional gate insight.

As the team discusses the best way to move forward, the snow sitting on top of the metal platform begins to glow. You and a team member quickly brush the snow to the sides of the platform to see what all the fuss is about. The two of you stand in shock as you realize the glowing is coming from a map that has been carved into the metal! Upon further inspection, you and your team member concluded that the glowing map is directing you toward an ominous cave opening in the mountain. At that same moment, your other three team members, who have been re-calibrating the gate finder tech, exclaim that the dimensional gate must be located within the mountain. With few options left, your team decides to head into the mountain opening.

Inside the cave, you find the remnants of an ancient Kree mining facility. You figure that this specific facility must have been abandoned a long time ago, mostly since it utilizes a spring system to launch the minecarts up to the main track. You flip the switch on the computer system that is attached to the spring and, to your surprise, it illuminates two graphs that describe the properties of the spring launching mechanism.

You and your team must use the only remaining minecart to launch yourselves across the oil-slicked straightaway of 10 meters, up a 13-meter long ramp (20-degree angle), and into the facility. You need to find out what your velocity will be at the top of the ramp.

You reach the top of the ramp and realize that there is no more track left and launch into oblivion. There is a 20-meter horizontal gap between the ramp launch and another platform 16 meters high. Where do you land horizontally on this new platform?