Maths Puzzle: How far does the fly fly?

Two boys set off on their bikes, at the same time, to meet each other.  They are 24 miles apart, and the road between them is perfectly straight.  The first boy cycles at 6 mph (he's only a very small boy), and the second boy cycles at 4 mph (because he's even smaller, and can't cycle as quickly).  A fly sets off from the first boy's handlebars at the same time as the boy starts cycling.  It flies in a straight line from the first boy's handlebars until it reaches the second boy's handlebars. It then turns around, and flies back to the first boy's bike, then when it reaches the first boy, it turns around again and back to the second boy's bike, and so on until the two boys meet.  The fly travels at 12 mph.

How far does the fly travel in total, from the moment the boys set off, until the moment they meet?  The time taken for the fly to turn around every time it reaches a boy can be ignored.

I like this type of puzzle - I liked it even more when I spotted an easy way of solving it. 

But first, the long way.

The fly sets off at 12 mph, at the same time as the second boy starts cycling towards it, at 4 mph.  There are 24 miles between them at this point, and the two travellers are approaching at 16 mph.  It will therefore take them


time = distance / speed = 24/16 = 1.5 hours


to meet.  This is the longest single part of the fly's journey, as the boys were at their furthest from each other.


However, in that time, the second boy has cycled 1.50 hrs x 4 mph = 6 miles, and the first boy has cycled 1.5hrs x 6 mph = 9 miles.
The fly has travelled 1.5 hours x 12 mph = 18 miles.  This makes sense - the second boy has cycled 6 miles, and the fly has travelled 18 miles, and 6 + 18 = 24 which was the starting distance between them.


Finally, while the fly has been flying the first leg of its journey, the two boys have reduced the distance between them from 40 miles to 24 - (9 + 6 miles) = 9 miles.

Now, the fly turns around, and starts to fly towards the first boy.  The first boy, remember, is cycling at 6 mph; the fly is still going at 12 mph, so their closing speed is 18 mph.  They have to travel 9 miles (the distance now remaining), so this will take:


time = distance / speed = 9 miles / 18 mph = 0.5 hours (which is 30 minutes).


In that time, the first boy travels 0.5 hours x 6 mph = 3 miles.
The second boy travels 0.5 hours x 4 mph = 2 miles.
And the fly travels 0.5 hours x 12 mph = 6 miles.


This makes sense - note that the first boy and the fly have together covered the nine remaining miles between them, 3 + 6 = 9


So far, the fly has travelled a total of 18 miles + 6 miles = 24 miles.


The distance between the two cyclists is now down to
24 - ((9 + 6) + (3 + 2)) = 4 miles

Continuing for a third leg, our tireless fly starts back from the first boy to the second boy.  The second boy is cycling at 4 mph, the fly is travelling at 12 mph.  Closing speed is 16 mph, and distance to cover is just 4 miles.


4 / 16 = 0.25 hours (15 minutes).


First boy covers 0.25 hours x 6 mph = 1.5 miles
Second boy covers 0.25 hours x 4 mph = 1 mile
Fly travels 0.25 hours x 12 mph = 3 miles


Distance remaining is 24 - (( 9 + 6 ) + (3 + 2) + (1.5 + 1)) = 1.5 miles
Fly has now travelled 18 + 6 + 3 = 27 miles

Clearly, this is going to take a long time to solve through this method.

Here's the shorter way.

The boys have to cover 24 miles.  The first boy cycles at 6 mph, and the second boy at 4 mph.  This means that they are approaching each other with a closing speed of 10 mph, and it will take them 24 miles / 10 mph = 2.4 hours (2 hours and 24 minutes) to complete their journey and meet up. Obviously, there are many different versions of this story, involving trains and so on, but the principle is the same (and you can change the numbers to make them more realistic - I think my boys are pedalling at walking speed!).


The fly, meanwhile, is flying at 12 mph.  This means that in the 2.4 hours it takes the boys to meet, it will fly 2.4 hrs x 12 mph = 28.8 miles.


It really is that easy.  No diagrams, no long complicated adding up then adding up some more.  Sometimes, all that's needed to solve a problem is a different perspective!

If you enjoyed this article, you may like some of my other, more recent posts on puzzles and games that can be solved with a calculator:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calculator Games: The Kaprekar Constant

More pseudoscience in TV adverts

"You use a toothbrush to clean your teeth, so why should your skin be any different?"

Because my teeth are a strong, tough, calcium phosphate compound designed for biting and grinding, while my skin is a soft, flexible material designed to be sensitive.

And so we begin again, although you probably shouldn't get me started on the strange and unintelligent things that advertisers say in TV adverts.  Too late, goes the cry.

Today, I have a little more on pseudoscience, inspired by a television advert I saw a couple of days ago.  I've already complained about the widespread use of hexagons in cosmetics adverts, but the commercial that I saw the other day reminded me of another considerable bug-bear of mine in the field of bad science:  the helix.

Two or three bands of various shades of brown spiral around a 'greatly magnified' strand of hair, before encapsulating it with a flash and a gleam of over-reflective sheen.  Why the helix?  Why the spiral?  I'm guessing that either it all comes down to DNA, or to the long-standing symbol for medicine? 

If it's a not-so-scientific rip-off of the DNA molecule, it might be worth mentioning that hair doesn't contain DNA.  No, despite what they may say on CSI, there's no DNA in hair.  The hair follicle at the base of the hair will contain DNA, but the hair itself won't.  So regenerating, recharging, repleneshing or otherwise repairing hair with DNA won't work.  No, sorry.

On the other hand, if the scientific message that we're meant to get is that the product is medicinal, then perhaps the helices should be snakes, as in the original symbol?  Probably not.  But there's no doubt that the helix is here to stay, at least for now, even if it only 'looks' scientific in the same way as a group of hexagons do. 

Until then, I think we'll have to keep washing our hair in icky goo, rinsing the dirt out of it with hot water and wait until somebody can find some chemically beneficial hexagons and helices; after all, they're all just using made-up names for chemicals.

Have I missed something?  Should I have added something else that keeps showing up in adverts in the name of science?  Let me know!

How about Glaxo Smithkline's woeful use of molecule diagrams in advertising?

Physics discussion: Escape Velocity

The story goes that Isaac Newton was sitting under an apple tree, when an apple fell on his head, and prompted him to wonder why it fell downwards, and not upwards or even sideways.  However, what history doesn’t tell us is that he probably got quite upset at having his afternoon nap interrupted by an apple, and, in his annoyance, threw the apple away as far as he could, declaring, “Stupid apples!”  He then wondered why the apple fell back to the earth, despite him throwing it away as hard as he could.

The same applies today (gravity hasn't changed much since then).  Consider throwing a tennis ball:  the harder you throw it, the higher it goes.  How about throwing it upwards, or even aiming for the moon (it’s not a million miles away, you know)? How fast does it have to be travelling, or how quickly do I have to throw it, so that it doesn’t come back down again?  We call this initial speed (how fast you have to throw it) the escape velocity.

Thinking in scientific terms, we can say that the apple (or the tennis ball) has escaped from the Earth’s gravitational pull, and will not fall back down to the earth.  It has maximum gravitational potential energy, and no kinetic energy (i.e. it stopped moving).  This happens at the edge of the gravitational field.

Since the kinetic energy at the start (i.e. from the throw) has all been converted into potential energy, we can say that the two are equal.

The potential energy is:

And the kinetic energy is:

where m₂ is the mass of the object being thrown, and m₁ is the mass of the Earth. 

 I’ll explain a bit more here about how this works, because at school I was taught that gravitational potential energy = mgh¸ where m is mass, g is acceleration due to gravity, and h is height – so that potential energy continues to increase with height.  So, when does PE = mgh stop being correct?  PE = mgh is not true when h becomes large, and g becomes very small.  The value of g changes with height; close to surface of the earth, mgh is an acceptable approximation, however at high altitude, g becomes

very much smaller.  It’s different at the top of a mountain than it is at sea level for instance.

So, the definition of potential energy is something else, it’s not mgh, it’s taken as something else.PE for all locations is equal to the formula given above.

Since we can equate these two energies, we have that:

Solving for this revised equation gives an expression for escape velocity, v, as:

Where m₂ is the mass of the Earth (in this case) and r is the Earth’s radius from centre to surface (i.e. from the centre of gravity to the point we’re launching from), since we have a bit of a head-start on gravity (we don't have to launch from the centre of the Earth).

Solving for all the numbers gives us an escape velocity of 11,181 metres per second, which is 34 times the speed of sound (Mach 34).  If you tried to throw an object at this speed, you'd probably either break your arm, or suffer friction burns from the air resistance as the air particles tried to move out of the way of your arm (and failed).  

It's also worth mentioning that I've not looked at air resistance, which at Mach 34 is considerable.  The sonic boom caused by the apple (or the tennis ball) would be extremely loud... in fact, I imagine the apple would turn into apple sauce, and the tennis ball would melt into a sticky, furry goo before it got anywhere near earth orbit.

I should explain at this point that escape velocity isn’t the speed that space rockets travel at when they take off.  This is really important.

An important point about escape velocity

Remember at the start that we were talking about throwing objects – where all the energy, and force is transferred to the object at the start of its flight.  With space rockets, the engines keep pushing the space rocket while it’s in flight, so they don’t have to travel as quickly, they just have to push upwards with a force that constantly exceeds their weight until they achieve an earth orbit.  This means that space shuttles, and space rockets, don't have to reach escape velocity.  Instead, they just have to keep pushing upwards with a force that is greater than their weight, until they reach an orbital height.

Other astronomy related articles you may find interesting:

Calculating the height of geostationary satellites
Calculating the angle of elevation of geostationary satellites
The Earth-Moon distance is increasing (published 1 April 2011)
What are constellations?
Calculating (and explaining) escape velocity

Astronomy 1: Stars, Planets and Moons

Following last night's visit to Keele Observatory, I thought it might be helpful to cover some of the basics of astronomy, and then move onto some more detailed topics.  Everybody's got to start somewhere, so I figure it's best to start with home, and move on from there.

The Earth spins on its own axis, taking one day to complete one revolution (one full turn).  This gives us day and night.

The Earth orbits (goes around) the Sun, going around the Sun in one year.  One year is 365.25 days.

Stars:  Stars are huge (very, very big) balls of gas that are carrying out nuclear reactions.  It might be easier to think of a star as an enormous nuclear reactor, constantly going out of control.  The Sun is a star.

Planets:  Planets are smaller balls of rock or gas that go around stars.  There are nine planets that go around our star, the Sun.  The nine planets are Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto (going in order from nearest to the Sun to furthest away).  

Moons:  Moons are smaller than planets, and go around planets in their own orbits.  Our Moon goes around the Earth in just under 28 days; some planets (such as Mercury and Venus) have no moons, while other planets (such as Jupiter and Saturn) have over 10 moons each.

One of the basic principles of astronomy is that smaller (lighter) objects go around larger (heavier) objects, and that's all due to gravity.  Galileo, who was one of the first people to make serious use of a telescope, saw Jupiter and four of its moons going around it, and started to wonder if the Earth goes around the Sun.  It wasn't a popular theory at the time, but a serious step forwards in our understanding of astronomy.

Our star, and its nine planets, are all part of a bigger group of stars (about 10 billion stars, roughly) that are all held together by gravity, in a group called a galaxy.  Our galaxy is called the Milky Way.  It's called the Milky Way because, if and when you can see the faint stars in our galaxy in the sky, they look like a milky cloud stretching across the sky.  Almost all of the stars that we can see in the night sky are in our galaxy.  Our nearest neighbouring galaxy is called Andromeda, and in the right conditions, it can be seen without a telescope or binoculars.

Why don't the planets crash into each other?
Because they're all going around the Sun at different distances.  Mercury is closest to the Sun, and completes one orbit in 88 Earth days, while Pluto, which is furthest away from the Sun, takes 220 times longer than the Earth to go around the Sun.

What is a light year?
A light year is a measurement of distance, and it's equal to the total distance that a ray of light would travel in a year.  The speed of light is 300,000,000 metres per second, or 186,000 miles per second, and there are 31 million seconds in a year (60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, 365.25 days in a year).  This means that in 31 million seconds, light would travel 9,467,280,000,000,000 metres, or 9,467,280,000,000 kilometres, and this distance is called a light year.  The distances in space are so far, that we need a meaningful measurement that we can use to compare distances between objects.  

The Sun's nearest neighbour is called Proxima Centauri ("proxima" meaning "close") and that's 4.22 light years away.  This means that light shining from Proxima Centauri takes just over four years to reach us, and that means that we're seeing what it looks like four years ago.  This, it is true, is a very strange situation, and that's because we're used to looking at objects that are much closer, where we can assume that we're seeing things as they are now (because the speed of light is very, very fast, and it takes fractions of a second for the light to travel from the object to our eyes).

I should make it clear that a light year (despite its name) is not a measurement of time, it's a measurement of distance!

In my next post, I'll try and move onto some more specific details, and answer a few questions that I've heard or been asked about astronomy.
Other astronomy related articles you may find interesting:
Calculating the height of geostationary satellites
Calculating the angle of elevation of geostationary satellites
The Earth-Moon distance is increasing (published 1 April 2011)
What are constellations?
Calculating (and explaining) escape velocity

Port Vale 1 Stevenage 3

I rarely watch football on the television, and it’s rarer still that I watch a live match; until yesterday, the last one was at least two years ago.  However, last night, as a result of my wife winning a prize draw, I went to see my local team, Port Vale play at home against Stevenage in a league fixture postponed from early December, when the pitch was frozen.

If you’re looking for a well-informed analysis of the game, I’d suggest looking elsewhere – I stopped watching football when it moved from terrestrial television to Sky, several years ago.  However, I can still make some comment on how the game developed, and what happened – call this an eyewitness account rather than a detailed commentary.

Stevenage started the game playing something like a 4-5-1 formation, with their number 10 playing pretty much as a lone striker against the Vale defence, and defended deep against Vale.  For the first 20 minutes or so, Vale had most of the possession, but were unable to do anything constructive with it; one long-range shot from outside the box, which went wide, was about the best they could muster.  It appeared that Stevenage were going to play for a goal-less draw, and it certainly looked as if they were going to hold Vale without much difficulty.  Vale lacked any sense of urgency or attacking strength, while Stevenage’s lone striker was having very little success against the Vale back line.

After half an hour, Vale seemed to have settled in possession, until one of the midfield players, possibly realising he had no clear way forwards, played a back pass from just outside the centre circle to the goalkeeper.  His pass was well weighted, but one of the Stevenage strikers fancied a look at it and started chasing it down, accompanied by his Vale marker.  In the end, however, they both gave up as the ball continued back to the keeper.  However, on its way back, it hit a divot about 10 yards from the goal line which took it slightly off course and straight past the unbalanced Vale goalkeeper into the net.

In almost all football matches I’ve watched, the moment the ball hits the back of the net is accompanied by a huge roar from the crowd and widespread celebration among the fans and players.  This goal was met by a stunned silence – from all the players and the crowd.  Stevenage brought around 100 fans, who were at the far end of the ground, and I don’t think they could quite believe what they’d just seen.  The Vale fans certainly couldn’t.

I was surprised that Vale didn't immediately start attacking with more enthusiasm, and seemed to settle back to their prior pattern of passing it around with no real sense of direction.  As the game progressed, however, they started to build an attack... much to their own downfall.  The ball broke along the Stevenage right wing, and their right back, number 2, chased it down, sending the Vale left back, number 15 Collins, back to cover.  However, the Stevenage defender beat his man and went charging down the right wing, delivering a good cross across the middle of the penalty area, until it went to the far post, and the resulting shot beat the advancing goal keeper.  The Stevenage players were very pleased with their first 'genuine' goal, and celebrated with some gusto.

The players were booed off the pitch at the end of the first half, and the highlight of the half-time break was an appearance by an ambulance who had to take one of the Vale fans off to hospital.  It was during the break that I started to understand the depth of feeling against the new manager, Jim Gannon.  The ambulance turned up; "Taxi for Gannon!" came the shout.  "Gannon should go," was another shout, and so on.

I should comment at this point on the referee and his interesting decisions.  Referees get some serious stick, it's true, especially from home fans whose team are losing (in my limited experience).  However, in my mostly impartial view, he didn't have a good match.  To quote another fan, "The referee's having a worse match than us!"  He didn't give any yellow cards in the first half, despite some hefty challenges, and eventually found his card in the second half.  He did miss some bookable offences for both sides (at one point, it was a good job he wasn't watching the slightly aggravated Vale defender who took a physical dislike to one of the strikers).

Vale started the second half with much more promise, substituting their number 7, Loft, who'd had a poor performance in the first half (he didn't make his passes well, didn't find much space and got beaten off the ball quite frequently) and bringing on somebody else (I wasn't paying attention, sorry).  However, they switched to a three-man attack with number 16, Haldane, moving onto the right wing.  The Vale midfielders and defenders gave him plenty of decent ball to run onto, and after a number of great runs against the Stevenage left back, he produced a cross from which Vale scored, via a rather fortuitous deflection (although the Vale fans would probably not have agreed).  Much roaring and clapping and chanting followed, and it looked as if Vale might retrieve the situation.

The game entered a lull shortly after Vale's goal, and Vale's players seemed to lose their energy, instead of repeating the process which had brought them their goal.  The Stevenage players suffered a number of minor injuries, much to the ire of the Vale fans, and some of the players too.  "Get him off;" "Get the stretcher on," and so on, were suggested for whichever Stevenage player had hurt his leg, or got a bit of cramp, or something.  I think the funniest was, "Can we get the ambulance back?"

During this lull, Stevenage got their third goal.  It followed a very delayed throw-in on the Stevenage right (the furthest corner from where I was sitting), and after a couple of quick passes across the Vale area, the Stevenage attackers managed to scramble the ball over the line in front of their travelling fans.

Haldane, who was Vale's brightest and most promising player, appeared to have taken a knock to his right foot shortly after creating Vale's goal, and despite trying to run it off, had to be substituted after about 70 minutes. To be honest, at that point, and after Stevenage's third goal, Vale's chances deteriorated.  There was plenty of stoppage time, due to the delays from the Stevenage injuries, but I didn't stop to watch it; the match was pretty much done after 80 minutes.  The most entertaining moment of the match came when one of the Stevenage players was substituted, and made a half-hearted attempt to shake the referee's hand - the referee in reply made a similarly half-hearted gesture.  "Hey up," came the shout, "Money's just changed hands there!"  And to be honest, the controversial bookings and non-bookings continued; the Vale defender Collins got a booking for a tackle which looked safe (after the referee had missed a worse challenge), and eventually the ref booked a total of three Stevenage players.

The funniest moment was the result of the fans' text-in competition to nominate the man of the match:  the goalkeeper won it; I think the Stevenage fans had been voting, because my vote would have been either Haldane, or Geoghan who had a pretty good game at the back.

Travelling on the surface of a star

As a follow-up to my last post calculating the distance to the Moon I was asked to calculate the following:

"Here is a question for you Mr Science. How long would it take a plane flyng at approximately 900km per hour across the surface of the biggest known star in our galaxy to travel full circle?"

Good question.  Let's assume that the star is perfectly spherical, which seems reasonable enough.  Now, to find the largest known star in our galaxy.  According to Wikipedia, the largest star in our galaxy is VY Canis Majoris found in the constellation Canis Major (meaning 'large dog').  It's a particularly bright star, which appears faint in the night sky because it's so far away.

VY Canis Majoris has a radius of 1800~2100 solar radii (it's 1800-2100 times wider than the Sun) - the figure varies as the star is surrounded by a nebula, which also makes it difficult to get an exact figure.  Taking 1800 solar radii as a minimum figure, to give us an approximate idea, this means that the star has a radius of:

1 solar radius = 695,500 km
1800 solar radii = 1.251 billion kilometres

Now, the circumference (i.e. the distance around the edge of the star) is 2 π r which gives a circumference of 7.866 billion kilometres.

And that's just the distance around the star's equator...  it's enormous.

Travelling at 900 km per hour, this would take 7.866 billion / 900 = 8.74 million hours.
8.74 million hours = 364,163 Earth days = 997 Earth years (assuming 365.25 days per year).

The exact figure depends on the value of solar radius, the rest is maths, but a round figure would be 1000 years.  Having said that, 900 km/h is not that fast - the speed of sound (Mach 1) is 1193 km/h.  The land speed record is held by Thrust SSC which achieved 1240 km/h in 1997, while Concorde used to reach 2170 km/h.

Still, doubling the speed from 900 km/h to 2170 km/h is only going to reduce the journey time to 500 years... so perhaps the question of time should be put aside.   The real question should be, if you're going to fly or travel on the surface of a star with a temperature of 3000 K, how are you going to keep the pilot flying, and stop him from frying?

Calculating the Earth-Moon distance

This post follows up my previous post on geostationary satellites.  Long before we were launching satellites (even non-geostationary ones), our natural satellite, the Moon, was orbiting the Earth.  As the moon goes around the earth, its phase (shape) changes, and in fact, the word "month" derives from "moonth", the time taken for the moon to go from new to full to new again.  This time is the time taken for one complete orbit around the Earth - the different phases of the moon are a result of us seeing a different amount of the lit half of the moon (I once based a very neat science lesson on this principle - in fact I used it in my interview lesson  and subsequently got the job).

One of my photographs of the moon, taken through a telescope.
The darkening at the bottom of the image is the edge of the
telescope's field of view

We can use physics, and our knowledge of the mass of the Earth, the value of pi and the time the Moon takes to complete one orbit, to work out how far it is from the Moon to the Earth.

Back to the two key equations that we'll need, which are the force on a body moving in a circular path:

where

And Newton's Law of Gravity

Equating the two, and rearranging to find r, gives us

This is the same equation used for geostationary satellites, and describes the basic relationship between the distance between two bodies (a planet and a moon, for example, or a star and a planet).  This gives it great power as it can be used in many different situations.

Turning to the current situation, then:

Calculation of the Earth-Moon distance:

G is the universal gravitational constant, 6.67300 × 10^-11 m³ kg^-1 s^-2

M is the mass of  the Earth, 5.9742 × 10²⁴ kg

T is the time to complete one orbit, which for the Moon is 27.32166 days, which is 2,360,591 seconds.

Plugging the numbers into the formula above gives the distance as 383,201 km

However, this is not the distance to the Moon from the Earth's surface.  Newton's law of gravity gives the distance between the centres of gravity of the two bodies.  I'm ignoring the radius of the Moon (which is perhaps an oversight on my part, you decide) but we must subtract the radius of the Earth from this value, to give the orbital height.  Radius of Earth = 6378.1 km, so the distance to the Moon is calculated as  = 376,823 km, or, if you prefer, (at 1.61 km to the mile), 234,147 miles.

Previously, I've learned that the Moon is about a quarter of a million miles away, so I'm glad the method I've used shows a figure which is 'about right' without any checking.  Looking at other sources, it looks like my figures are close enough, considering the assumptions I've made.  One key assumption I've made is to suggest that the moon travels in a circular orbit, and it doesn't.  It has an elliptical orbit, which means the distance from Earth to Moon changes during the orbit - so I've calculated an average distance.  Still, my figure is pretty close, and not a million miles away (and next time somebody reliably informs you that their opinion is not a million miles away, you can tell them that not even the Moon is that far away).

Other astronomy related articles you may find interesting:

Calculating the height of geostationary satellites
Calculating the angle of elevation of geostationary satellites
The Earth-Moon distance is increasing (published 1 April 2011)
What are constellations?

Calculating (and explaining) escape velocity
Measuring g using a pendulum