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Challenge Questions

Stop in and exercise your brain. Talk about this month's Challenge from Specs & Techs or similar puzzles.

So do you have a Challenge Question that could stump the community? Then submit the question with the "correct" answer and we'll post it. If it's really good, we may even roll it up to Specs & Techs. You'll be famous!

Answers to Challenge Questions appear by the last Tuesday of the month.

Pack Your Bags: Newsletter Challenge (October 2015)

Posted October 01, 2015 12:00 AM
Pathfinder Tags: challenge questions

This month's Challenge Question: Specs & Techs from IHS Engineering360:

You are a regular hiker. When you pack your backpack you decide to put the heavier items close to the bottom of the backpack and the lighter, less dense items near the top. Is this the best way to pack? Is this scheme good for your body?

The answer to this challenge will be posted later this month, right here on CR4.

43 comments; last comment on 10/08/2015
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Mars Farms: Newsletter Challenge (September 2015)

Posted September 01, 2015 12:00 AM
Pathfinder Tags: challenge question

This month's Challenge Question: Specs & Techs from IHS Engineering360:

A group of settlers are debating possible locations for a Mars colony. All proposed settlement sites are located in the northern hemisphere. When asked why this is, the settlers responded "For the longer growing season." Why would one hemisphere of Mars have a longer growing season than the other?

And the answer is:

Mars has approximately the same tilt as the Earth (Mars = 25° vs Earth=23.5°). So just like Earth, Mars has seasons depending upon its position in its orbit. However, compared to Earth, Mars has a very elliptical orbit (Mars = 0.094, Earth=0.017). That means the length of the seasons vary more than on Earth due to changes in orbital speed as Mars orbits the Sun. In the northern hemisphere of Mars, the seasons are as follows:

Spring - 7 months
Summer - 6 months
Fall - 5.3 months
Winter - 4 months

The situation is reversed in the southern hemisphere of Mars where Spring lasts 5.3 months, Summer lasts 4 months, Fall lasts 7 months and Winter lasts 6 months.

Thus the growing season is longer in the northern hemisphere of Mars.

73 comments; last comment on 09/30/2015
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Coin Flips for Cash: Newsletter Challenge (August 2015)

Posted August 01, 2015 12:00 AM
Pathfinder Tags: challenge question

This month's Challenge Question: Specs & Techs from IHS Engineering360:

Try this game: you flip a coin until you get a head and you win one dollar every time you flip the coin. For example if you get a heads the first time, you get one dollar; if you get a tails the first time and a head the second time, you get two dollars; and so on. What is the maximum expected dollars you can make every time you play this game?

And the answer is:

Every time you flip the coin the probability of getting heads is ½, so the probability of winning just one dollar is ½; the probability of winning two dollar is 2/4 (= ½) and so forth. Then, the expected winning is given by this series:

This is a convergent series so its sum is a finite number. You can find the value of this series using normal mathematical techniques, or you can use the following C++ program to find that the series converges to 2.0. So you will win no more than $2.00 each time you play the game.

// Solution to Challenge Question for August 2015

#include <iostream>

#include <math.h>

using namespace std;

const int MAX = 100;

int main ( ) {

int n; // n is the number of coins you end up flipping

double expectedValue = 0.0;

for (int i=1; i <= MAX; i++) expectedValue += i/pow (2,i);

cout << "\nExpected Average Value = " << expectedValue;

return 0;


22 comments; last comment on 08/25/2015
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Apollo 11 Communications Log- Gimbal Lock: Newsletter Challenge (July 2015)

Posted July 01, 2015 12:00 AM
Pathfinder Tags: challenge questions NASA

This month's Challenge Question: Specs & Techs from IHS Engineering360:

CapCom - "Columbia, Houston. We noticed you are maneuvering very close to gimbal lock. We suggest you move back away. Over." Michael Collins (CMP) - "Yes. I am going around it...How about sending me a fourth gimbal for Christmas?"

Why did Michael Collins ask for a fourth gimbal for Christmas?

And the answer is:

The full communication log can be found here:
In the discussion above, mission control (CapCom) is communicating to Michael Collins who is piloting the Apollo Command/Service Module, that he is nearing gimbal lock. This is in reference to the Inertial Measurement Unit (IMU) which has Outer, Middle, and Inner gimbals (see diagram below). The outer gimbal is mounted on the navigation base which in turn is rigidly mounted to the spacecraft.

Michael Collins had flown in Gemini 10, which had a fourth or "redundant" gimbal. This redundant gimbal was mounted outside the normal outer gimbal. Normal operation would then be to use the inner three gimbals to drive the stabilizing gyro error signals to zero while the fourth gimbal could be used to keep the middle gimbal near zero and away from the gimbal lock orientation.

The general lesson is, rotations in 3D in space are tricky. That is why 3D simulators are usually based on quaternions rather than Euler Angles. Quaternions have a built-in extra degree of freedom that avoids gimbal lock. Essentially adding a fourth (redundant) gimbal does the same thing.

11 comments; last comment on 07/21/2015
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Airbag Physics: Newsletter Challenge (June 2015)

Posted June 01, 2015 12:00 AM
Pathfinder Tags: challenge question

This month's Challenge Question: Specs & Techs from IHS Engineering360:

This month we had the largest automotive recall in U.S. history: 33.8 million vehicles made by 11 different automakers were recalled due to malfunctioning frontal airbags. These airbags were made by the parts supplier Takata. So, let's think about airbags for this month's challenge question.

We know that airbags are very soft inflatable devices; during a crash the airbag is the best accessory to save a life. The question this month is: Why? Why do airbags save lives? It is not enough to say that lives are saved during a crash because the airbags are soft devices. I am looking here at the physics that supports this notion.

And the answer is:

Let's assume, for instance, that a car is moving at 60 mph and crashes into a tree. A passenger in the front seat, before the crash, is moving at 60 mph. At the time of the crash their body will stop (normally by hitting the dash of the car, if there is no airbag). So the passenger momentum will change from mv to 0, where v = 60 mph.

Now, in order to change momentum, according to Newton, a force needs to act for a given amount of time. This concept is a consequence of Newton's second law, which is stated as follows

F = ma

where a is the acceleration of an object of mass m and F is the force generated by this acceleration. We know that acceleration is a change in velocity/time, so we can write the above expression as follows

and finally,

The expression on the left is called the impulse. So the impulse is equal to the change in momentum. t is the time we must apply the force F in order to produce a change in momentum.

The equation can be written as,

In the case of the crash in a car without airbags, at the moment of the crash the passenger will start moving at 60 mph toward the dash. At the very moment that the passenger hits the dash, a force F will be applied by the dash to the passenger (Newton's first law) and in practically zero time (because the dash does not subside) the speed of the passenger goes to zero. Looking at the last equation, we see that the force the dash is applying to the passenger is almost infinity. This, of course, will kill the passenger.

If airbags are available, the passenger will hit the airbag instead of the dash. As soon as the passenger hits the airbag, a force F will be applied to the passenger by the bag, but because the airbag is flexible this force will be applied during a finite time (not zero). This is the time measured from the moment the passenger touches the bag until the speed of the passenger is zero. Because the time is not zero, we see from the above equation that the force is not infinity.

Therefore airbags save lives because they allow the force used to stop the speed of the passenger to act for a certain period of time.

27 comments; last comment on 06/06/2015
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Styrofoam Slide (May 2015)

Posted May 01, 2015 12:00 AM
Pathfinder Tags: challenge question

This month's Challenge Question: Specs & Techs from IHS Engineering360:

Try this experiment: slide a Styrofoam cup full of water across a smooth (finished) wood surface. Make sure the speed is around 10 cm/s. You will notice that at this speed droplets of water from the cup shoot up to about 20 cm. Explain why.

And the answer is:

As you slide the cup at this speed, energy couples into the sliding cup. The almost-flat bottom of the cup slips and sticks to the surface, producing nonlinear vibrations. This is exactly a relaxation oscillator (producing a non-sinusoidal repetitive signal) that immediately produces standing waves in the surface of water. As you continue pushing the cup the water crests of the standing wave increase in height, causing drops of water to break from the surface and be projected into the air.

49 comments; last comment on 06/27/2015
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