July 4, 2011

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## Can a Formula 1 car drive upside down? Can an ordinary car fly?

February 9, 2010

The 2010 Grand Prix is about to hit Melbourne next month so this week, mathspigs we are going to look at car aerodynamics. There are two factors in car aerodynamics. Lift(L) and Downforce(D)

LIFT (L)

Car designers didn’t worry about aerodynamics until the sixties when cars could go fast enough to experience aerodynamic lift, which naturally, reduced tyre traction.  Skirts are added to cars to not only reduce this lift but also create a ‘suction’ (Bernoulli Effect”) under the car.

Here is the Chevy ’69 Camaro from Camaro Untold Secrets.  At 115 mph ( 185 kph which is not only fast but over street speed limits!!!!!) it experienced a front Lift (L) of 375 lbs or pound force.

We’ll simplify the aerodynamics and assume this graph is a parabola as follows.

NB: the only units to use in this equation are L (lb force) and v mph.

Once you know this equation you can calculate the velocity required for a ’69 Camaro to leave the ground due to front Lift (Assume once the nose lifts the downforce no longer applies ie. equals zero.)

We know the weight of the ’69 Camaro = 3,675 lb.

So mathspigs if the ’69 Camaro can go fast enough so that the front Lift equals its weight, it will fly!  At what speed will ’69 Camaro fly?

Downforce (D)

The ultimate design in car aerodynamics is the F1 racing car.  The Downforce (D) on the car is created by the front and rear wings, which work in reverse to aeroplane wings pushing down on the car) and the underbody gap which crates ‘suction’ by the Bernoulli effect.

Williams FW31 weight including driver  605 kg

Renaut R30 total weight 605 kg

Toyota TF 109    weight 605 kg

BMW Sauder C29    620 Kg

Ferrari F60    605 kg

Downforce is approx:

35% rear wing

25% front wing

40% defuser on underbody

While downforce equations are complex ( See Wikipedia ) they approximate to the following equation for a standard F1 car design:

Could the Downfoce (D) on a F1 car exceed its weight so that it could drive upside along a … very long tunnel? I’m thankful to New Scientist (Aerial Glue 20 Jan 2010) for this insight.

Now mathspigs you have the mass of F1 car above so you can calculate the speed at which a typical F1 racing car could drive on the roof of a tunnel glued to the ceiling by its own downforce.

NOTE: This equation only works for a limited range of car mass, m. You can’t reduce m by dumping bits of the car. Likewise, if you add bits to the car to increase m eventually the aerodynamic constant, k, will change.

Does the weight of a driver make a difference?

Mathspig did stand in the pits in the Australian Grand Prix in Adelaide in the ninties ( with a borrowed ticket) not far from Ayrton Senna (Right. Tragically killed in 1994) and I was shocked to see how small he was. His height is recored at 171 cm but that is Mathspig’s height and he was shorter.  Today while some F1 drivers are over 6 ft (183 cm) including Australia’s Mark Webber (Right) most are not tall and, therefore, not that heavy.

When the KERS (Kinetic Energy Recovery System), which stores braking energy, was introduced 2009 drivers lost weight to accommodate system.

Nico Rosberg 76 kg to 66 kg

Kubica 78 kg to 70 kg

Heidfeld  56 kg to 59 kg

While Trolli, Hamilton and Vettel reduced their weights to 64, 67 and 62.5 kg respectively.  More info

Typically, a driver will lose 5 kg just competing in a Grand Prix.  IF the driver dropped 15 kg would that effect the upside down speed of an F1 racing car?  Drop the mass (m) value above and see if it makes much of a difference.

Or, instead, if your School Principal was driving the F1 racing car ( Assume an average  F1 driver weighs 65kg. How much heavier is your Principal to an F1 driver? Guess.) Increase the mass (m) in the above equation by this amount and you can calculate the speed  at which your school principal and F1 car would drop off the tunnel ceiling.

Meanwhile, even Mythbusters hasn’t tested this stick to the ceiling theory. Why?

Even the slightest bump on the ceiling could disrupt the underbody ‘suction’ and down goes the \$zillion car and driver upside down and at speed.

This hasn’t stopped some Youtube mockups of F1 cars driving on the ceiling!!!

## How Maths Solved a Real Murder!

October 1, 2009

It is the aim of Mathspig to show you maths as it is used in the real world.  As this is a REAL murder conviction we will be very serious in our comments.

A woman was found dead at the bottom of The Gap at Sydney’s Watson Bay in 1995. It wasn’t until 1997 that University of Sydney physicist, Rod Cross, was asked if the victim could have jumped off the cliff.  In 1998, the coroner declared an open finding in the death of Byrne.

It wasn’t until 2003, however, that the police contacted Ross to check the maths. He said that Caroline Byrne couldn’t have slipped or jumped.  The case was reopened and in 2006 Byrne’s ex-boyfriend Gordon Wood was arrested in London and eventually found guilty of her murder. Wood was sentenced last year to 17 years in jail with a non-parole period of 13 years.

Why we are interested in this case, mathspigs, is because Cross, The Physicist, made the comment when asked during the trial (See The Australian Wednes 30th Sept 2009 ) that the maths involved was not rocket science but maths high school students would be able to master. Can we?

Here is some information you might need:

Height of cliff = 29m later found to be 25.4m

Distance from base of cliff (of body) = 11.8 m

Weight of body = 57 kg

We will assume  influences of weather or air resistance are not significant. We will not include the sloping rock surface at the bottom of the cliff in the calculations.

So mathspigs, you can do it? And keep in mind this is a real case. It’s not Numb3rs or NCIS.

You know y (Height of the cliff) so you can calculate x (Position of body from cliff) for different velocities (Vx) of a person leaving the cliff. There was a limitation on the speed of someone running off the cliff as there was only a 3 – 4 m lead up to the cliff edge. Here are some known running speeds:

Speed of fastest man on earth = 10.4 m/sec       Usain Bolt ,Jamaica  running 100m in 9.58 sec (@  37.4 kph)

Speed of fastest woman on earth = 9.29 m/sec   Evelyn Ashford, USA running 100m in 10.76 sec ( @ 33.4 kph)

Speed of average sprinter = 19 – 24 kph

These calculations alone did not bring about the conviction. The prosecution then had to prove that a person could be thrown at sufficient speed to land 11.8m from the cliff base. To do this Cross re-enacted the experiment with two policemen and a policewoman of similar size to the victim. The policewoman was thrown into an Olympic Pool. The policemen could achieve an exit velocity of the policewoman of 4.15 m/sec, 4.37 m/sec and eventually using the strongest thrower 4.85 m/sec.

Could your maths put a murderer behind bars, mathspigs?

For more information see Rod Cross’s book, Evidence for Murder: How Physics convicted a Killer published next month. ( New South Books  )

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# Wood Aquitted

On the 3rd DEC 2008 Gordon Wood was sentenced to 13 years jail for the 1995 cliff top murder of Caroline Byrne.

On 24th FEB 2012 Wood was acquitted on the murder charge in an unanimous decision by three Appeal Court judges. Justice McClellan said Wood’s behaviour around the time of the death had “raised suspicions”. The prosecution case was based on evidence given by Associate Professor Cross, but the Court of Appeal labelled his evidence biased as he had later written a book titled Evidence for Murder: How Physics Convicted a Killer. Justice McClellan noted in his judgement that “suicide could not be ruled out” and that “suspicion, even a high level of suspicion, was not enough” to convict someone of murder.

Of course, Caroline Byrne, may have been murdered. The question that remains unanswered is ‘By whom?’

## What are maths teachers for, sir?

July 17, 2009

A major fast food firm, all right, McDonald’s has put the entire Australian Year 7 – 12 Maths curriculum on line and it is FREE!!! #mce_temp_url#

I never thought I’d be enthusing over McAlgebra or McTrigonometry or McX, but this site has been recommended to me by maths teachers and it looks great. Students can view worked examples of maths problems, hear the problems solved step-by-step, work at their own pace and even track their progress through the website.

If the entire maths curriculum can be accessed for free on line what ARE maths teachers for?

Gather round mathspigs and I will tell you why we need maths teachers.

Maths is boooooring for the majority of students. It needs a maths teacher to breath fire into the concepts to make them interesting, relevant and exciting. Otherwise, students feel maths is on a par with re-translating the dead scrolls.

Mathspig’s mission is to provide maths teachers with fun, pop-culture based, media-savvy maths to excite students about the whole idea of maths. These topics can be used by maths teachers in the same way retailers use ‘loss leaders’. Get them in the door and you can sell ‘em anything…. almost!!!!!!

Since maths teachers often moan about students not knowing their tables… here is a COOL MATHS GAME called

MATH LINES : X-FACTOR which is all about using your knowledge of tables …quickly.  #mce_temp_url#

This is how I would use it in the classroom. Get this game up on the Smartboard and call kids up one at a time to have a go!!!!!! It’s fun.

And it is FREE  x FREE = FREE !!!!!!!!