Brain Teaser No 1

Two-sheds,

Many thanks for your advice.

I am playing around with various bits of software including one called 'Imaging' and have saved a 45KB image in JPEG/TIFF. But the Forum poster won't accept this type of file.

I don't seem to have MS Paint. I still use Windows 95 and the 'Office 2000' versions of Word/Excel/Access/Powerpoint etc

Will keep trying

Cheers

Don

I recommend you scan at a reasonably high resolution and then resize the image in software prior to compression. jpg seems most appropriate for colour images. Many of the larger images seen on the forum will be hosted elsewhere so the size limit doesn't matter, you can also display multiple images in a single post.

Paul

Boy Racer....

Last night, coming home, I stopped at traffic lights.

When I started off, I decided to time the car to the next set of lights which I knew were exactly 2,145m away. (don't ask!).

I accelerated uniformly for 15 secs, then drove at a steady speed before decelerating uniformly for 30 secs, to a halt. I covered the whole 2,145m in 2 minutes exactly !

The length of road has a 50mph speed limit. Did I break the limit, or not.

Cheers

Don

Go Faster....

Most cars have a foot-operated throttle...

And when you bang it to the floor boards (push it forwards) the car goes faster....some more so than others.

Now I teach people to fly and we all start on light aeroplanes, Cessna, Piper, Grumman etc

These are piston engined, propellor driven and have hand-operated throttles that you push forward to increase engine rpm and power. ie when you push the throttle forwards, the engine goes faster, just like in your car....

BUT, the aeroplane itself won't go faster, in fact it will actually slow down, if all you do is push the throttle forwards....

Reason(s)....?

Cheers

Don

Boy Racer

Don,
Let's get back to some interesting problems, rather than the basic A-Level physics questions...
John

Whilst not a massively talented mathematician, I enjoy the problems - albeit finding the solutions as interesting as the questions.

I say that the "boy racer" problem is achievable without breaking the speed limit.

My reasoning (albeit loosely) is that in the 2 minutes you would be able to travel 2.667 km at a constant 50 mph (80km/h).

Given that the overall distance travelled is less than this, it should allow for acceleration and deceleration within the speed constraint.

Sorry if there are more words than numbers !

John,

Boy Racer

Don,
Let's get back to some interesting problems, rather than the basic A-Level physics questions...

"Ouch!!"

Cheers

Don

PS see below

L-T-D and James,

As you will have guessed by now, the Boy Racer is a straight forward A-Level maths question, without any twists or turns. (dreadful pun). In fact, it probably wouldn't make it onto a real A-Level paper!

Despite the improbable looking numbers, you can calculate the steady speed of the car and its a nice stisfying number, so you know its right when you've got it.

Of course, if you didn't do A-Level maths, or, like me, you're a bit rusty. it can seem a bit of a brain teaser. Which it why I slipped in the 50 mph bit, rather than simply asking for the steady speed.....

So, A-Level maths solution = boring but precise speed calculation
Trial & Error solution = tedious, but you should get the exact speed eventually

Apologies for letting the standards drop.

Cheers

Don

James,

Because you also have to change the pitch of the prop(s)?

nice try.........but wrong, I'm afraid.

This is a characteristic of most, if not all, light aeroplanes and quite a few big ones.

Most of the light ones have a fixed-pitch propellor.

If the engine is very powerfull, and "sensibly" positioned in the airframe, the characteristic might not be exhibited.

Cheers

Don

John,

First, I sympathise and fully recognise your request to get back to 'real' brain teasers, rather than the routine, thinly disguised A-Level questions.

And I will.

BTW, setting questions isn't a perogative exclusive to myself......(John, Paul, Dan, Steved, L-T-D, James...)

However, I believe a wide range of teasers isn't amiss on this forum....we're not all ex rocket scientists like Paul R and co.. One or two contributors have actually expressed delight at being simply able to understand the less demanding questions, never mind being able to supply any answers....

Even some of the 'best' teasers here, are little more than A-Level questions dressed up and carefully worded.

BTW, the "throttle" one isn't on any A-Level sylabus,, although I anticipate your cry that it is purely a 'knowledge' thing, if not quite 'general knowledge'.....again I stand "guilty as charged" and only slightly repentant.

But I hope you will continue to visit this thread, even if only to watch in quiet amusement as we struggle with our desire to pass A-Level maths again......and again.....and again...

Cheers

Don

Throttle

The forces acting on an aeroplane are complex, but the situation can be simplified to :-

Lift, acting more or less upwards through the centre of lift
Weight, acting downwards through the centre of gravity
Thrust, acting more or less forwards more or less along the axis of the crankshaft and propellor
Drag, acting backwards, but very difficult to even gues where it lies
Plus some controlling forces acting on the elevators, ailerons, rudder and trim tabs.

Of course, none of the forces remains constant for very long (weight decreases as fuel is used) nor do they act in fixed directions or position (the drag moves quite a bit when the flaps are moved)

In general, the principal forces do act as couples Lift & Weight; Thrust & Drag

Draw a picture of a high-wing a/c (eg a Hurcules C130) and try positioning the principal forces. This might help you sort out why light a/c slow down when the thrust power is increased.

Cheers

Don

Odds and Evens

Prove that the product of two odd numbers must be odd.

You can assume that the sum of two even numbers is even and you can assume that the product of two even numbers is even.

and probably, quite a few other assumptions have to made as well......
like the sum of an even number and an odd number is odd
and the product of an even number and an odd number must be odd

Cheers

Don

PS. on reflection, given all the assumptions, you might just as well assume that the product of two odd numbers is odd, but why not have a go anyway....

quote:

---

Originally posted by Don Atkinson:

Draw a picture of a high-wing a/c (eg a Hurcules C130) and try positioning the principal forces. This might help you sort out why light a/c slow down when the thrust power is increased.

---

I'll take a stab at this if I may: assuming the aircraft was in straight and level flight, increasing thrust power intensifies the airflow over the wings which, being aerofoil surfaces, therefore induce lift causing the aircraft to nose up into the climb. The climb would then lead to a decrease in airspeed.

Am I in the ball-park, Don?

JonR

quote:

---

Prove that the product of two odd numbers must be odd.

---

Don,

It's early on a Monday morning, but here's my logic:

If we assume the opposite, and can show the statement is false, then the product must be odd. Given the product has a unique prime factorization, and 2 must at least one of the factors, then at least one of the terms on the LHS must have 2 has a factor. But since it has 2 as a factor it is even -- a contradiction.

cheers

Dan

It can't be this simple can it?
If 2x is even, then 2x+1 is odd. So
(2n+1)(2m+1) = 4nm+2m+2n+1
The first three terms of the expansion are even therefore the product is odd.

Simon

JonR,

Am I in the ball-park, Don?

Well.....more like the right parish, rather than ball park, but generally....yes

Propeller air -flow does generate some lift but not an awful lot. It tends to affect the rudder and elevators more and the effect these have on control.

But, where you are right, is that the aeroplane will start to climb when you increase power.

You just need to identify the main reason for this climb and you're well on the way to a good answer

Cheers

Don

Dan,

Nice logic.

Not my solution, but none-the-less sound

Cheers

Don

Simon,

It can't be this simple can it?

....errrrr, yes it can.....

Well done.

Hope you're not going to join Mr Channing and complain that it was too easy/boring/routine/O-level maths......OK I know it was, but still, well done

Cheers

Don

Planes
Is it that the thrust vector does not pass through the CoG nor the CoL? I haven’t worked out which it is or is it the relationship between the two – a bit like boat stability.
Increasing thrust will cause the plane to rotate, and either pitch up if the thrust is below or down if the thrust is above the relevant centre. If it pitches up (which is what I think will happen for a small high wing plane) then angle of the attack and the drag will both increase causing the plane to climb and slow down.

Simon

Simon,

You're getting there.

Planes are generally designed to be stable and behave safely.

The lift/weight and thrust/drag vectors are not co-linear but they do act as couples and each couple is designed to be able to balance the other couple (with a little help from the tail-plane/elevator).

Also, if power were lost, as in engine failure, it's better to have the nose drop and let the a/c enter a glide, rather than pitch up and risk a stall.

If you draw the principal vectors you will probably see why the nose pitches up when you add power. This in turn causes the climb and the speed drop.

Low-wing a/c are also designed like this, but it's easier to visualise with a high-wing a/c.

In practice, to increase the speed, you need to both increase the power AND prevent the nose pitching upwards by easing the c/c forward. You also need to prevent yaw/roll developing from the increased propellor airflow and torque reaction etc etc....

Cheers

Don

Ahhh… I used to jump out of planes (a lot) This explains why pilots were so worried about CoG – if you move too much weight aft and change the direction of rotation of the weight-lift couple, you are screwed –since air speed is close to stall anyway and increasing thrust in this situation just makes things worse. (High wing single engine aircraft)

Simon

The throttle thing

Hopefully the attached picture makes things clearer.

The force exerted by the propulsion device acts below the force exerted by drag. So for normal flight a upward force is required on the tail to correct for this.

If the force exerted by the engine increases then if this force also acts below the center of gravity the plane will see an increase in this upward turning torque and to compensate increase upward force is required on the tail to prevent the nose lifting.

Matthew

...so the key for plane design is that the line of force of the engines acts below the center of gravity and that the drag of the plane tends to be slightly below (though not to far) the center of gravity, so as to prevent stalling in case of engine failure.

Matthew,

Thrust (engine/propellor/jet) acts forwards (more or less); Drag acts backwards (more or less).

The aim is to have the centre of drag above the the thrust line.

Then, when power is reduced there is a tendancy for the nose to drop

Also, Weight acts downwards; Lift acts upwards (more or less)

The aim is to have the centre of lift behind the centre of gravity (weight).

Then again, there is a tendancy for the nose to drop when the power is reduced.

Increse the power (thrust) and the thrust/drag couple increases and causes the nose to pitch up. This happens imediately and before inertia can be overcome to increase the speed.

Cheers

Don

A fighter test-pilot is flying at 30,000 ft, rolling along at nearly Mach 2, and
his control tells him to press button A which opens the small bomb bay doors. A
few seconds later after he confirms that's been done he's told to press button B
which releases the new type of bomb fixed in the bay.
Everything works 100%
correctly, and the bomb weighing some 150 lbs is definately on board, only the
bugger doesn't drop, even the the release mechanisms etc, etc, all function
perfectly, WHY ?

Pish: This is an oldie, and I definately didn't get it right, and about 100 odd
people I'd asked over the years didn't get it either, all except some snotty
nosed little Turkish lad (who was 9) and sussued it in no time at all at all,
innit.No doubt this is already on this thread somewhere, and most of you know
the answer anyway, even if you do try it with someone who doesn't, or try it
yourself if you don't know it? You know it makes sense.

Fritz Von Capsten !