(Physics - Medium) Escape from Earth

You are the lead project designer for Space X's new moon orbital mission. The first step is simple enough; Get off Earth.

Getting off of Earth means achieving an escape velocity that allows you to fly away from Earth without using anymore fuel. The kinetic energy of this launch must at least match the Earth's gravitational potential energy to achieve escape velocity.

What is the escape velocity required to break free of Earth's gravity? Assume air resistance is negligible.

Table of constants:
Mass of the earth, M  = $5.976\times 10^{24}\ kg$
Mass of shuttle and all, m = $28,833\ kg$
Gravitational constant G = $6.67x10^{−11}\ \frac {Nm^2}{kg^2}$
Radius of the earth, r = $6378\ km$

(Materials - Medium) Steel Hanging From The Ceiling

Part 2 of 2 - Let's build on yesterday's problem.

Why do we bother finding out how much something deforms, when it's stress that gives us a measure of whether something will fail?

In the real world, directly measuring stress can be pretty difficult. It's much easier to stick a strain gauge to the interested area of a part to measure the amount something deforms. From there we can use an equation to relate strain to stress. We go into this more in today's solution (after you try solving the problem of course!).

You hang a steel cylinder from the ceiling. At 3 various points, you decide to hang some heavy stuff. Young's Modulus of steel is 210 GPa.

How much will the length of the steel cylinder change?

(Materials - Easy) Tug of War Over an Aluminum Rod...

You and your friend are playing tug-of-war over an aluminum rod that you each want for yourself.

The diameter of this rod is 25mm, and the standard elasticity of aluminum is 69 GPa.

How much are you guys elongating the rod by fighting over it?

(Logic - Med) Mooooooooooo

In life it’s often not about getting the “answer” to a problem, but rather knowing how to solve it. When we see a complex issue that we don’t have the answer to, but yet we know what method to apply… IT FEELS AWESOME. We often hear people give solutions, but rarely do we see HOW they got there. Maybe we just assume “man, that guy’s smart!” But in reality, they got there via a certain way of thinking; and that can be learned.

So today I propose something new. I propose we lay our minds on the table to teach each other “how to think differently”… We all seek to become better 'problem solvers.' But LEARNerds alone cannot accomplish this. Rather, it takes a community. Below is a question borrowed from the Harvard Consulting Club. It’s interesting since there’s no right answer, but instead many potential methods of solving. Take a look. Give yourself 5 minutes to think about it. And then explain (in the comments) HOW you’d go about solving!

QUESTION: A fast food chain recently bought a meat-processing outlet to supply it with fresh hamburgers and other meats. The shop process is: Cows enter from one end of the shop, meat gets processed in the middle, and then the meat gets packaged and delivered at the other end.

The manager of the butcher shop however cannot decide whether to have the cows walk or run into the meat processing room. How would you go about helping him figure out what to do?

(NOTE: Since there’s no way for us to hide others’ responses, we ask that you please honorably refrain from reading others’ answers until AFTER you comment. There are no wrong answers.)

(Math - Medium) Volume of a hypercube

Kudos go out to LEARNerds user Mitch for suggesting today's question!

You and a friend are discussing math in multiple-dimensions, like true nerds! Then she brings up the tesseract...

This is a stereoscopic image. It can be viewed in 3D by crossing your eyes. See how

A tesseract is a 4 dimensional cube. It is as if you took a 3 dimensional cube and extruded it into the 4th dimension, same as you would take a 2 dimensional square and extrude it into the 3rd dimension to form a cube. It is also called a hypercube.

The conversation becomes a debate when someone asks, "What is the volume of a hypercube?"

Assuming the length of one side is $a$, what is the volume of a hypercube?
We'll assume you two mean the 3D volume.