Brain Teaser No 1

I think it matters not whether the fly is 'stationary' and the car hits it, or the car is stationary and the fly moving, or both are moving.

We have no conceptual difficulty with dropping a ball onto the ground and it bouncing. At some point all parts of the ball have had a velocity of '0' wrt the ground. If we happened to be observing this from a moving lift would it surprise us?

I haven't answered the question, today I'm not convinced there is one.

Paul

I haven't answered the question, today I'm not convinced there is one.

There is a question, but it is sometimes difficult to recognise it. The answer, on the other hand can seem relatively trivial, although I personnaly don't think its trivial at all.

Your bouncing ball helps to identify the question. But because the ground is stationary. we have no problem in accepting the ball is also stationary during the time it reverses direction.

Now, if that ball had been thrown horizontally at an express train, it would still reverse direction, and in doing so, would, for a short time, be stationary. so, two things are in contact, (train and ball). One is stationary for an instant, (ball) the other should be stationary for that same instant, (train) but clearly it isn't.

What is really going on?

Cheers

Don

The logic in the statement is flawed. The fact that the fly and car are in contact when the fly reaches an instantaneous velocity of zero does not imply the same is true of the car.
John

SPLAT !

I have always felt this question was a bit like the Tortoise and the Hare.

You know, the one that the Greeks argued about for ages, some trying to PROVE that the hare could never overtake the tortoise, when it was plain to see by experiment that the hare never had any problem in overtaking the old slowcoach.

I have wondered what is going on when a cricket ball is sent flying 180 deg back by a cricket bat, and also what is going on when one atom (of a fly ?) gets hit by a train (say).

Cheers

Don

I think phrasing the question in terms of a pair of parachutists, one of whose parachutes fails to open, is more interesting.

Jumper A leaps, chute opens, floats happily. Jumper B leaps, chute doesn't open, speeds past A, hit ground and starts returning toward A. So obviously was at '0' relative to A for an instant. Therefore either A stopped falling instantaneously or the Earth leapt away from him.

Paul

This is the gist of my first thoughts about the fly and car.. they are conditioned because I read an article in New Scientist..... they do align a fair bit with John's thoughts.

A fly does stop a car..... but not the whole car, just part of the small local area where the fly makes contact, and then not for very long.

All objects, no matter how rigid they seem, are flexible to some extent. So the car’s windscreen, on being struck by the fly, deflects backwards very slightly. That small piece of car not only stops for a short period, but can actually move backwards. It takes considerable force to do this (glass being fairly rigid) but it should be remembered that the forces involved in any type of impact are typically quite large.

The force exerted by the fly on the car is the same size as the force exerted by the car on the fly...a large force. And such a force acting on the small mass of the fly gives rise to a very large rate of acceleration. In fact, the rate of acceleration of the fly is so great that it accelerates up to the speed of the car in only the short distance by which the windscreen has been deflected.

Having got the fly up to speed, the windscreen then springs back to its original shape. Because it springs back very quickly the deformed part actually overshoots its original position and a vibration is then set up as it springs back and forth trying to regain its original form. This probably gives rise to the sound we hear when the fly hits the windscreen.

This simple picture is complicated by factors such as the crushing of the fly’s body and inertia effects in the glass, but it does demonstrate some of the principles that I think might be involved.

Now, there are other considerations, but I will post them later....

Cheers

Don

Most of you who have contributed to the "Splat" teaser have suggested or indicated that there is a fundamental concept at large. Paul didn't consider there was even a question to answer.

So, is there a philosophical rather than a physical question/answer lurking? Is there a paradox at large?

Around about 450 BC Zeno of Elea posed a paradox. A moving object is always in motion (remember, in 450 BC we haven't quite got to Einstein and relativity).... "A moving object is always in motion, and yet at any given time it is somewhere, ie it is stationary."

One of our problems, even today, is that we can't see, measure, or really imagine an infinitely small time, any more than we can imagine infinity. And we probably never will!

Cheers

Don

Driving technique

I was driving home late one night last week, when I realised that I was low on petrol. I wasn't sure I could reach the next motorway service station, about 50 miles away.

What is the best way to drive in order to conserve fuel and maximise the range of a car? It was about 10.30pm and the motorway was clear.

I have a 4.2 litre fuel-injected automatic.

Would a different technique apply to a 1600cc, carburettor fed, 5-speed manual?

Cheers

Don

quote:

---

Originally posted by Don Atkinson:
"... and yet at any given time it is somewhere, ie it is stationary."

---

Heisenberg might not agree with that.

Dan

Heisenberg might not agree with that

Dan, who was/is Heisenberg ?

Cheers

Don

Don,

I'm sure you're just playing along, but a quick google found: AIP link
I was of course referring to the uncertainty principle:
The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.
Or, to paraphrase your statement, at any certain time it is nowhere.

cheers,

Dan

Dan,

I had genuinely forgotten about Heisenberg and his work otherwise I would have said..."I am a bit uncertain as to who Heisenberg is..." I had to look him up in the encyclopeadia.

Thanks

But he does help us understand the dilema faced by us (well, me at least) and the Greeks etc with the concept of motion and being in one place.

So, what happens to the car when the fly splats into it? Momentum before and after is relatively plain, (if you'll forgive the pun) but during the splat....?

Cheers

Don

So, what happens to the car when the fly splats into it? Momentum before and after is relatively plain, (if you'll forgive the pun) but during the splat....?

Don,

I haven't really had time to ponder your question fully, but if by "when" you mean "the instant," then momentum is undetermined. If one considers any finite amount of time, then momentum is approximately conserved if you neglect losses to heat and sound, and gains from changes in gravitational potential energy of the bug-car system.

This reminds me of my first encounter with calculus and such things as "lim x->0." I recall questioning how a line of zero length can have a slope.

Dan

Hi Cliff,

This post has been difficult to draft and I know I haven't managed to get the tone right, so I do apologise in advance for that. I am also probably being far too sensitive, but I thought I detected a slight touch of concern in your last contribution. Let me start with a very sincere thanks for your contributions to this and other teasers; and this is genuine.

The thread is open to anybody to set teasers and although I have set most, the best have been set by others such as Bam (ladder; tethered goat, monty hall) and ken c (various circles). Also Omer set a few (probability questions) and Paul Ranson and Dan M have answered more that their fair share. And your contributions have been spot on as well. (My apologies to all those other contributors over the past couple of years whom I haven't specifically mentioned.) I hope you (and others) will all continue.

This last pair of teasers were posted because I felt they needed a bit of careful thinking. I accept my role also includes playing 'devil's advocate' and sometimes this gets close to 'flogging a dead horse' so again, my apologies. (BTW, I've occasionally wondered whether 'flogging' refers to 'selling' a dead horse or 'hitting' a dead horse).

The need for careful thinking (IMHO) revolved around the paradox (if we can call it that ?) of "one body (fly) being stationary at some point in time (is it ?) yet at that same instant (how small is an instant ?) being in contact with another body (car) that is clearly in steady motion (is it really steady ?) relative to an observer". How do we resolve this apparent paradox, or can it all simply be explained by small (elastic ?) deformations in the car and large (plastic ?) deformations in the fly?

I appreciate that your first and second response to the Splat question, were aligned very closely to my physical, Newtonian understanding of what was happening. I also accept that the question failed to provide precise physical properties of the bodies involved. So we can only generalise the effect, or we have to talk about a wider range of possibilities.

I accept its not one of the better teasers in this thread, but observe that it has prompted a variety of responses from Paul R (no question to answer) through yourself to Dan M who opened my eyes again to Heisenberg and his uncertainty principle.

I hope my replies to various respondents haven't touched any raw nerves. The one that triggered your last reply was obviously aimed at Dan who was beginning to probe into what I see as the interesting philosophical area of 'infinitely small' and 'infinitely large', which I feel are related to "how did it all start and how will it all end"....and I mean before the "big bang" etc

Jeepers, what a load of waffle. But I've typed it now and its sincere, so I'll post it anyway...

Cheers

Don

Hummm, I didn't think my was response was philosopical at all.

-Dan

I didn't think my was response was philosopical at all.

Compared to the Greeks; and if read by Einstein; it could be almost pragmatic.......

Cheers

Don

Mountain Air

Today, one of my students asked,

"If hot air rises, why then is it colder at the top of a mountain than in the valley below ?"

What would your explanation cover?

Cheers

Don

Don,

I believe that the flogging of a dead horse refers to hitting it.
If a farmer wanted a workhorse to pull something he'd hit it's rump to get it going, and a jockey would use the whip to urge his mount to greater speed.
Not much point in doing either of those if the poor animal is deceased, hence the saying.

Glad to finally answer a question in this thread!

Regards,
Steve.

Hi Steve,

Glad to finally answer a question in this thread!

Welcome to the forum and this thread....and well done!

Cheers

Don

Cliff,

Boyle.

He said he'd vaguely remembered a Robert Boyle from his school days who had done some work on gas pressures and temperatures. So he'd looked up Boyle in a cheap computer encyclodaedia to find

Boyle's law states that under conditions of constant temperature, the pressure and volume of a gas are inversely proportional.

So he felt the "constant temperature" bit was a bit of a problem.....

although he sensed there was something in what Boyle had to say, but couldn't quite figure it out..

Cheers

Don

Pressure drops as you go up the mountain. .....Well according to my Casio barometer watch anyway.

Cliff, it looks like you were the only one willing to give this a shot.....

....and not bad neither. (I know, it sounds patronising.....sorry, it isn't meant to be)

After Boyle, messers Lussac and Lussac came along with further properties of gasses and eventually ISTR we ended up with a relationaship

P1*V1/T1 = P2*V2/T2 = Konstant for any given gas

So if the pressure drops, which it does as you climb (your Casio....) there is a tendancy for the temperature to drop as well.

But the volume does change as well...

Adiabatic Cooling

If the pressure on a quantity of air is reduced (e.g. by the ascent of a mass of warm air), expansion takes place and the air cools adiabatically, i.e. without loss of heat to its surroundings.

The next post will include a longer background......

Cheers

Don

The slightly longer version...

Radiation

The earth receives its heat from the sun in the form of electromagnetic waves or radiation. The short heat waves responsible for the transmission of the sun’s heat through space resemble light and radio waves in many of their properties. They pass easily through the atmosphere and heat the surface of the earth, but they have little direct heating effect on the air itself.

The increase in temperature, which the heat waves produce at the earth’s surface, depends on the latitude of the place, the time of day, the season of the year and the heat capacity of the surface layers. For example, forests and water require more heat to raise their temperatures than rocks and sand.

The hot surfaces re-radiate heat waves of longer wave-length than the original heat waves from the sun, and the long heat waves are absorbed and reflected back to earth by clouds and water vapour in the atmosphere. On a clear night these heat waves escape into space, so that clear nights are often cold and frosty.

Convection

Air in contact with the warm earth is heated and expands. The expansion causes a decrease in density, and the warm air, being lighter than the surrounding cold air, begins to rise while cold air flows in to take its place. The rising currents of warm air - convection currents - carry the heat from the earth’s surface to the bulk of the atmosphere. Notice that the air is heated from below, and consequently temperature decreases with increasing altitude. Warm up-currents, cold down-currents and horizontal winds form a highly complex and ever-changing convection system over the whole surface of the earth.

Density

Air being a compressible substance, the weight of the upper parts of the atmosphere compresses the lower layers so that we find a maximum density of 0.077 lb. per cu ft. at the surface and a decrease of density with increase of height. Although at first glance it would seem that the lower temperature at altitudes would cause an increase of density, this is only a minor effect, which is completely masked by the loss of density caused by lower pressures at altitudes.

Adiabatic Cooling

It the pressure on a quantity of air is reduced (e.g. by the ascent of a mass of warm air), expansion takes place and the air cools adiabatically, i.e. without loss of beat to its surroundings. (N.B. The fall in temperature is caused by the expansion of the gas against the enclosing pressure of the surrounding air.)

Likewise, a mass of air, which descends to a lower level, is compressed and heated adiabatically, i.e. without gain of heat from its surroundings.

If the air contains some water vapour, continuation of the adiabatic cooling by expansion will cause it to become saturated, for cold air is saturated by a smaller amount of vapour than warm air. The air has been cooled to its Dew Point Temperature and further cooling causes a cloud of water droplets to appear.

Have you ever noticed that the barrel of a bicycle pump gets quite hot if you pump up a tyre quickly? The air in the pump is heated adiabatically by the sudden compression, which occurs during the downward strokes of the plunger.

Lapse Rate

The average rate of decrease of temperature in the lower levels of the atmosphere is 1.6 deg C. per 1000 ft. increase of altitude. This is known as the normal lapse rate, but the lapse rate for any particular day may depart very considerably from this value.

Cheers

Don

Hot air....

Of course the mountain rocks themselves tend to heat up as well, but this tends to get masked by expanding (and adiabatically cooling) air rising and cooling etc...or wind blowing and rising over mountains and in doing so cooling on the upwind side....

Cheers

Don

Squares, plants and planes

The other day I was gardening. I had 13 plants to set out in one bed and 26 to plant out in another bed. As I was working away, I realised that 13 can be expressed as the sum of two squares (4 + 9) and so can 26. Hmmmmmmm.

The next day I was flying from Blackbushe to Leicester, a distance of 78 nautical miles. I had marked the 1/3 and 2/3 distances on my map because they were very close to easily identifiable landmarks. Having passed the 26 nm mark (1/3 point) after 13 minutes flying, I realised that I had 52 miles to go with an estimated flying time to destination of 26 minutes. Now, I don't know what made me realise it, but 52 can also be expressed as the sum of two squares. Interesting.

So I wondered what was happening here. It looks like if I have a number n that happens to be the sum of two squares ie n = a^2 + b^2, then 2n must also be the sum of two squares.

Can you prove this ?

Cheers

Don

2 is the sum of two squares, 1*1 + 1*1. 2*2 is 4 which isn't the sum of two squares.

So I assume the question relates to sums of different squares?

There are some interesting (to me at least) infinite thoughts here. Every integer has a square, there are an infinite number of sums of two different squares, but no more than the infinity of integers.

The assertion is that there are the same number of even sums of two squares as all sums of two squares. And there's nothing wrong with this assertion on the face of it. I think.

Paul