Passy's World of Mathematics

Rocket Science and Cars

Posted on October 8, 2013 by Passy

Image Source: http://image.popularhotrodding.com

This lesson is a quick look at Rocket Science and Cars.

It is all about Acceleration.

A normal car engine provides steady straight line acceleration for the car to overtake other vehicles.

There is a particular steepness or gradient to the graph.

However, Rocket Engines provide massive acceleration for the large heavy rocket to be able to shoot up fast and escape the earth’s gravitational field.

So what happens when we attach a Rocket Engine to a Car ?

Well this has been done by the Mythbusters TV Show, as well as right here in Australia, during the filming of the original “Mad Max” Movie.

Myth Busters Jato Rocket Car

In this myth, an airforce pilot supposedly attached a rocket engine onto his car, with powerful tragic results.

The following videos show the Myth Busters episode about this.

What kind of Mathematics was involved?

– Calculating the power from the rockets and estimating how fast the car might go

– Calculating the correct geometry for mounting the rockets onto the car

– Calculating the geometry of the rocket mounting assembly, and the strength of the building materials needed, so it would be strong enough
eg. use steel rather than wood, and cross members for strength

– Working out the geometry of an internal cage with triangulation for strength to mount the remote control assembly

– Fine tuning and calibrating the remote control devices to drive the car like a human would

– Realign the centre of gravity of the car by attaching a heavy front bumper bar to counterbalance the weight of the rockets on the back

– Designing a special sized steering wheel on the remote controls to give the correct required radius of turn inside the car

– Estimating the Probability of how dangerous it would be to have a human occupant, and working out that the test car would definitely have to be remote controlled

– Estimating the Probability for all possible disastrous and dangerous events occurring, and taking appropriate safety precautions

– Working out where cameras would need to be positioned so that none of the action was lost

– Working out the size of a large enough flat area to run the experiment

Here are the videos of the Mythbusters episode.

Part 1 Video: – 15mins

Part 2 Video – 15mins

Although the myth was busted, the Mythbusters did one final additional test, involving building a ramp to use as part of the experiment.

Part 3 Video – Add a Ramp 13 mins

Mad Max Rocket Car

Image Source: http://ih1.redbubble.net

The original “Mad Max” movie made right here in Australia over 20 years ago, involved attaching a rocket to one of the cars for a particular scene.

The results were similar to the Mythbusters experience, in that the rocket provided too much power and made the car uncontrollable.

The Nightrider’s spectacular crash was the result of a military booster rocket being installed in the back of the car.

Passy worked as a Military Rocket Scientist a few years later, with some of the guys involved with building the rocket motor for the Mad Max Movie.

Although many mathematics calculations and tests had been performed, it was always suspected that the end result would be dangerous and unpredictable.

During the filming, the rocket mounted car went out of control, missed the target fuel tanker, and veered off into the field where it chased the film crew for a 1/4 mile.

The on-camera explosion was a later re-creation using a safer towed car.

The two sequences were then joined together to make the crash sequence that is in the Mad Max movie.

At the end of this YouTube video you can see where the Military Rocket fires off and the Nightrider’s car crashes.

Note that the movie clip is PG Rated, there does not appear to be any bad language, killing, or grotesque violence.

Image Source: http://content6.flixster.com

There is a Great Trivia Page on the original Mad Max Movie at the following link:

http://www.imdb.com/title/tt0079501/trivia

There is a new Mad Max Movie that should be out in 2014, and Mel Gibson is listed as being in the cast.

Well that’s it for our action packed Mathematics in Rocket Science lesson.

If you want to learn all about Rocket Science, then the “Midnight Tutor” has an excellent set of videos on the following YouTube Channel:

http://www.youtube.com/user/midnighttutor/videos

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Midpoint Between Two Points

Posted on October 2, 2013 by Passy

The 90 mile straight section of highway lies between the towns of Balladonia and Caiguna in the Australian Outback.

Some friends of ours recently drove this stretch of the Eyre Highway.

If their car started making some strange noises along this section of road, would they be better off heading back to Balladonia, or continuing onto Caiguna?

Well assuming there is a mechanic in each town, it would depend on whether they had crossed the halfway point, (or “Midpoint”), of their journey.

Eg. If they had not passed the halfway midpoint, then they are better off heading back to Balladonia, because it is less distance to drive with a dodgy sounding engine.

This lesson is all about finding midpoints on straight lines.

Finding the Midpoint along the “90 mile Straight” is fairly simple.

Finding the Midpoint Between Two Points

To find the Midpoint Between any two (x,y) coordinates, we simply add the x coordinates and divide by 2, and also add the y-coordinates and divide by 2.

Here is an example where we have worked out the Midpoint manually.

Try the following example, and see how you go.

(The answer is further down this web page).

Here is the answer to the above Example Question.

Practical Application of Midpoint

Image Source: http://www.dirtybutton.com

If a plane flying between two cities gets some engine damage, it would need to be determined from the Coordinates in the GPS Navigation system, if the plane is less than halfway through its flight distance or not.

A decision can then be made either to turn back, or keep flying to the final destination.

The Pilots are not going to get out graph paper and count squares to work out where they are !

Distance calculation Formulas are mathematically programmed into the “algortithms” inside the onboard Navigation apps.

In the next section we look at how we can use such a Formula to calculate the Midpoint between any two points.

Midpoint Formula

The following is the Midpoint Formula for two points.

Here is a quick three and a half minute video about how the Midpoint Formula is used.

Midpoint Formula – Working Out Steps

To obtain the Midpoint we follow these steps:

Step 1 – Label our two points (x1, y1) and (x2,y2)

Step 2 – Get the average of the x values, then get the average of the y values

Step 3 – Simplify our x and y answers and write them as an (x,y) coordinate.

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Posted in Straight Line Graphs | Tagged cartesian midpoint, how to find midpoint, how to work out the midpoint, middle point between points, midpoint between cooridnates, midpoint between points, midpoint calculator, midpoint for two points, midpoint formula, midpoint video, midpoint worksheet, real world midpoints | Leave a comment

Gradient and Slope

Posted on September 30, 2013 by Passy

Anyone who has ever ridden a bike uphill knows what a “Gradient” or “Slope” is all about !

But in Norway there is a clever way to save the hardship of big hills.

It’s called a “Bicycle Lift”, and is in Trondheim Norway.

Image Source: http://weburbanist.com

It is the world’s only functional bicycle lift, designed to help bike riders ascend the hill.

Image Source: http://weburbanist.com

The device is called a “Trampe” and opened in 1993. The hill it is on goes from the town center to the University. The “Trampe” has also become a major tourist attraction.

Basically you purchase tokens, then strap your foot into the escalator, and it pulls you up the hill.

Image Source: http://weburbanist.com

For the smoothest ascent the advice is to keep the right leg extended behind the body, shift the body weight to the footplate, and lean slightly forward on the bike.

So there is some weight distribution and balancing mathematics involved with the entire process.

Image Source: http://weburbanist.com

As can be seen in the diagram a continuous cable loop powers the lift, and it is driven along using electric motors with gears.

Definition of Gradient

The Mathematical value of a Gradient or Slope involves comparing how far up we have gone, against how far across.

Original Image Purchased from Photozone.com

Types of Gradient

There are four types of Gradient: Uphill, Downhill, Flat Horizontal, and Straight Up Vertical.

Mathematically we name these : Positive, Negative, Zero, and Infinite.

POSITIVE Gradients and Slopes go UPHILL from left to right.

Original Image Purchased from Photozone.com

NEGATIVE Gradients and Slopes go DOWNHILL from left to right.

Original Image Purchased from Photozone.com

ZERO Gradients and Slopes are FLAT or horizontal and do not go up or down.

Original Image Purchased from Photozone.com

INFINITE or UNDEFINED Gradient or Slope goes straight up, and is neither uphill or downhill.

This final type of Gradient or Slope is one which goes straight up.

A bike cannot be ridden vertically, but it can be stored in this position.

Because it cannot be ridden vertically, we call the slope or gradient “UNDEFINED”.

original image from http://www.thehappybike.com

Introductory Video About Gradient

The following video gives a really good overview of what Gradient is, and then does a basic straight line gradient question.

We highly recommend watching this video.

Gradient on Cartesian Plane

Numeric Values for Gradients are usually worked out using measurements between points that are located on a reference grid.

Usually this Reference Grid is an X-Y “Cartesian Plane”, and the points are (x,y) coordinates on this grid.

For our previous example, the Steepness or Gradient or Slope of the line connecting points A and B is 8/4 = 2.

We say the size of the uphill slope between the two points is “2”.

Because the direction of the line is uphill, the gradient is positive 2.

To calculate Gradient or Slope, we always draw a right angled triangle around our two points.

We then count “up” squares, and “across” squares.

There are examples on how to do this later in this lesson.

Note that there is also a “Formula” way of working out gradient or slope from a pair of (x,y) points, but this is covered in a separate lesson.

In this lesson we are only looking at finding gradient or slope by manually counting squares on an X-Y Grid.

Related Items

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Posted in Gradient, Straight Line Graphs | Tagged angle of elevation, bicycle lift, bike escalator, bike lift, bike tow, gradient, gradient calculator, gradient counting squares, gradient in real world, gradient maker, gradient of line, gradient online quiz, gradient videos, gradient worksheets, incline, norway bike lift, online gradient slope quiz, parallel lines gradient, parallel lines slope, perpendicular lines gradient, perpendicular lines slope, real life gradient, slope, slope calculator, slope of line, slope videos, slope worksheets, Straight Line Two Points Calculator, types of gradient, types of slope, why gradient is called m, why is gradient called m, why is slope called m | 5 Comments

Mountain Gradients

Posted on September 26, 2013 by Passy

Image Source: http://www.seanasmith.com

Cable trains are called “Funiculars”, and are used to go up and down very steep incline Gradients.

These gradients are too steep to have a train push or pull the carriages, and so the carriages are pulled up and down by steel cables.

The steepest Gradient cable train in the world is actually right here in Australia, at Scenic World in the Blue Mountains.

The train runs at an angle of 52 degrees, which is steeper than any normal train or road vehicle could ever go.

Here is a great two minute video about the Australian Cable train at the Blue Mountains.

Definition of Gradient

Gradient represents how steep a slope is in terms of its vertical change over a given horizontal Distance.

The Gradient is expressed as the Rise / Run

or as m = Vertical Change / Horizontal Change

A Gradient of m = 3/2 means that for every two meters we go horizontally, our hill goes up by 3 meters.

The Angle of Incline can be found by doing Inverse Tan-1 (3/2) which comes out as 56 degrees.

56 Degrees, or 3/2 is quite a Steep Gradient.

Fenicular Trains

Image Source: http://u.slimg.com

Many Feniculars are powered by large diesel or electric motors, but the Stanserhorn Funicular in Switzerland is “self-powered” and requires no energy.

It is the perfect eco-friendly form of naturally powered transport.

How the Swiss train works under its own power is by having a continuous cable connecting a carriage going up with a carriage going down.

When the “down carriage” is at the bottom of the mountain, the “up carriage” is at its stopping point further up the mountain.

The force of Gravity on the train carriage going down the mountain, pulls the other train carriage that is going up the mountain.

Along the way there are passing sections.

These are located at a midpoint where it is exactly known the up train will meet the down train.

Image Source: http://www.funimag.com

Here is a photo of the two cable trains passing each other.

Image Source: http://imagebanana.com

There are also “Funicular” self powered gradient cable trains in other parts of the world, such as the funicular which was built in 2005 along the Zambezi River, and is called the “Victoria Falls Eco-Lift”.

Image Source: http://www.funimag.com

If you are interested in Feniculars, then check out the following website, which has lots of Feniculars featured from all around the world:

http://www.funimag.com

Mountain Cable Car Gondolas

Image Source: http://4.bp.blogspot.com

On the Stanserhorn Mountain, the Fenicular Train can only go so far up the Mountain.

At the train’s finishing point, the gradient of the mountain slope is too big for safe train travel.

Passengers get off the train, and board a cable car which goes to the top of the mountain.

By stringing the Gondolas of the Cable Car above the ground from towers of varying heights, the slope of the cables can be kept at an approximate 45 degree value where the cable cars can operate.

Image Source: http://flickr.com

Stanserhorn Videos

Here is a video which shows the Stanserhorn Fenicular journey.

It is a long eleven minute video which shows the whole trip.

We suggest watching the 5 mins to 5 mins 30 secs part where the two trains meet; and then watch from 9 minutes 20 seconds to 10 minutes 30 seconds part where a train comes into the mid mountain station to meet up with the cable car.

This second video shows the cable car ride section of the journey to the top of the mountain.

Be warned that it does have an interesting swiss folk song musical backing track.

Finally, if you are mega-fit, you can enter the race to the top of the Stanserhorn, which is shown in the following video.

The Race is conducted at the height of summer, when there is no snow on the mountain.

Finally, here is a good five minute video about why trains have trouble going up any type of hill.

Mountain Vegetation and Temperature Gradients

The air temperature is warm at the base of the mountain with the sun shining and blue sky, but very cold with ice and snow at the top of the mountain.

There is a decreasing gradient as we go up the mountain with temperature progressively getting lower.

The Vegetation and Soil also has a Gradient Change as we go up the mountain.

The vegetation and soil decreases follows along with the temperature decrease as we go up the mountain.

In the lower section where the train starts, there are houses, grasslands, flowers, and deciduous trees.

There is plenty of rich soil, and lush pastures for cows and other livestock.

Image Source: http://amazonnaws.com

However, as we ascend the mountain, the Gradient becomes steeper, there is less vegetation, with rocks amongst the grass, and only evergreen pine trees.

The vegetation change is due to the decrease in temperature, and the increase in rockiness, (less soil and more rock and gravel), as we go higher up the mountain.

Image Source: http://www.funimag.com

As we get right up near the top of the mountain, the sides are very steep, there is virtually no soil, and the only plants which can survive in the cold snow are pine trees.

Image Source: http://tripadvisor.com

As we get close to the top where it is very cold, only a few pine trees manage to cling to the steep sides of the mountain.

Image Source: http://tripadvisor.com

The top of the mountain has a great view of the Swiss Alps, but the temperature is a lot colder than where the sun is shining on the lake at the base of the mountain.

Image Source: http://tripadvisor.com

Passy has been up the Stanserhorn Mountain in Switzerland, and it is a unique gradient experience which is highly recommended!

Ascending any big mountain, there are the gradients of the mountain slopes, the temperature gradient, and the soil and vegetation gradients.

Changes in these real world gradients make for a unique and scenic journey from the lakeside town of Luzern, up to one of the highest peaks in Switzerland.

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Posted in Gradient, Straight Line Graphs | Tagged angle of incline, climate gradient, fenicular, fenicular mathematics, gradient formula, gradient in real world, incline, mountain gradients, mountain mathematics, mountain temperature gradient, real world gradients, scenic world railway, slope, slope formula, stanserhorn fenicular, train gradients, vegetation gradient, why gradient has letter m, why gradient is important, why is m used for gradient, why m is ued for slope, why m stands for slope | 5 Comments

Introduction to Probability

Posted on September 21, 2013 by Passy

Image Source: http://cdn.caradvice.com.au

Statistically the most likely age group to have an auto accident is the 18-25 year old male group.

(They are young inexperienced drivers, and for some reason they don’t always seem to drive as carefully as other people do).

These young guys have the highest probability (or chance) of having an auto accident.

So a Car Insurance Company would be crazy to insure only 18-25 year old males as their Policy Holders.

Instead they need to “spread their risk”, and insure a range of drivers, including young people, as well as older people with good driving records.

Image Source: http://cdn.rsvlts.com

Likewise, an Insurance Company offering Fire Insurance would soon go broke, if they only insured people living in high “wild fire” danger areas in California, or dangerous “Bushfire” areas of Australia.

Such areas have a high probability or chance of fire damaging peoples properties.

Instead Insurance Companies need to have policy holders living in a variety of locations.

This way the premiums from the people living in areas where there are hardly ever fires, can cover the payouts for victims of high fire risk areas.

Insurance Companies use Probability, Statistics, and other advanced Mathematics to assess and cover risks. The experts who do this work are very highly paid people called “Actuaries”.

Introductory Videos on Probability

Watch these videos to learn about Probability.

This first video is a one minute Video about Probability being a value between 0 and 1.

This next video goes into more detail about Probability, and shows some examples of what the chances are of various things occurring.

The following video is a bit out there, but summarises most of the main concepts associated with Probability in less than ten minutes.

This video also covers combination proabilities which we will cover in a later lesson.

We apologize in advance if some people find the word “crap” offensive, or do not like some of the bizarre scenarios in this guy’s word problems.

Overall the Video is probably PG Rated.

The Video Word Problems are covered very quickly, and so you may need to pause and rewind the Video in some places.

Probability Summary

Here is a summary of the key points we need to know in relation to Probability.

Probability of 0% or 0 = cannot occur = impossible

eg. The sun rising in the West.

Probability of 100% = Definitely will happen

eg. Ice on your bare skin will feel cold.

The chance of any event happening is always in the range of 0% to 100%, or 0 to 1 if dealing in Fractions.

A 50% or 0.5 or 1/2 chance occurs for getting heads when you toss a two sided coin that only has a head on one side.

A 50% chance is not definite, and is only an approximation or estimate or average.

Eg. You could toss a coin 10 times and only get heads 3 times.

But if thousands of people all tossed a coin ten times, we would expect that the majority of them would get 5 heads and 5 tails.

There is a “Probability Scale” which looks like this:

Image Source: http://ictedusrv.cumbria.ac.uk

The Probability Formula for an even happening is as follows:

Pr(Event) = outcomes you want to get / total possible outcomes

If two independant events are involved, in an “AND” situation, then multiply the separate probabilities to get the answer.

Eg. For two coins tossed and there is heads on the first coin AND heads on the second coin:

Pr (HH) = P(HH) = 1/2 x 1/2 = 1/4

For one event “OR” the other one occurring, add the probabilities to get the answer.

Eg. Probability of Throwing a normal six sided dice and getting a 1 OR a 6 = 1/6 + 1/6 = 2/6 = 1/3

Introductory Notes and Examples on Probability

The following two web pages have some great introductory notes and examples about Probability:

http://www.mathsisfun.com/data/probability.html

and

http://www.mathgoodies.com/lessons/vol6/intro_probability.html

The Math Goodies page, even has interactive spinners, and dice rollers on it, that you can click and make events happen.

Dice Simulators

For a great simple free online dice roller, which allows the user to roll between 1 and 60 dice at a time, click the following link:

http://www.random.org/dice/

The following freee online app has several dice simulations; but the sound effects can be a little annoying.

Click the link below to use this simulator

http://www.curriculumbits.com/prodimages/details/maths/mat0005.html

This dice simulator allows you to adjust the number of sides on the dice for interesting investigations.

Click the link below to use this simulator

http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks1/maths/dice/

Coin Simulator

This simulator does coins and dice, and you can choose what you want to do, for example: two dice.

As you spin the coins it also keeps a tally of your results.

Click the link below to use this simulator

http://www.subtangent.com/maths/coindice.php

Another Two Coins Simulator, where you can pick various world currency coins to use, can be found here:

http://www.random.org/coins/

Interactive Spinner

The following interactive allows players to spin the colored spinner, and it tallies the results as we go, provided that we click on the Results Button at the bottom of the screen.

Click the link below to use this simulator

http://www.mathplayground.com/probability.html

Probability Worksheets

Here are some worksheets that can be done to practice Probability Skills.

Click here for Probability Worksheet 1

Click here for Probability Worksheet 2

There are two similar worksheets with answers at the following PDF link:

Click here for Probability Worksheet 3

Virtual School Videos on Probability

The “Virtual School” YouTube channel has an excellent set of videos on Probability.

Here are the main videos, which are all well worth watching.

Video about Mutually Exclusive Events

Video about Calculating Probability

Listing Probabilities and Sample Spaces Video

Probability Multiplication Rule

Example 1 of Multiplication Rule

Example 2 of Multiplication Rule

The General Addition Rule

Probability – Tree Diagrams

The following two videos show how to use Tree Diagrams to answer Probability Questions.

Tree Diagrams can be used when we have Independant Events – the first event does not effect the second event.

Examples include tossing a coin twice, rolling a dice several times, or picking lollies or candies from a bag several times.

Part 1 Video – 6 minutes

Part 2 Video – 5 minutes

In this second video we look at sampling without replacement.

Probability Games

Here are some fun online games to play and practice Probability.

Carnival Probability Game

In this game, students learn the practical application of the concept of probability by playing carnival games.

The object is to win as many tickets as possible.

Click the following link to go to this game:

http://mrnussbaum.com/probfair-play/

Colored Balls Machine Probability Game

Click the following link to play this game:

http://www.kidsmathgamesonline.com/numbers/probability.html

Higher or Lower Card Game

Click the following link to play this game:

http://www.subtangent.com/maths/higher-lower.php

Zoo Animals Probability Race Game

This game is a little bit out there, but is good fun once you get into it.

Click the following link to play this game:

http://www.bbc.co.uk/bitesize/ks2/maths/data/probability/play/

More Online Probability Activities

There are lots of Games, Interactive Lessons, and Test Yourself Online activities at Mrs MacIntosh’s webpage:

http://www.tvdsb.ca/webpages/cmacintosh/mathematics.cfm?subpage=193944

There are plenty more Online Games and Interactives at the the Johnnies webpage:

http://jmathpage.com/JIMSProbabilitypage.html

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