WHEN WE MOVE ROUND THE SUN FASTER... Physics can be entertainment

WHEN WE MOVE ROUND THE SUN FASTER

Paris newspapers once carried an ad offering a cheap and pleasant

way of travelling for the price of 25 centimes. Several simpletons

mailed this sum. Each received a letter of the following

content:

"Sir, rest at peace in bed and remember that the earth turns. At the

49th parallel that of Paris you travel more than 25,000 km a day.

Should you want a nice view, draw your curtain aside and admire the

starry sky."

The man who sent these letters was found and tried for fraud. The

story goes that after quietly listening to the verdict and paying the

fine demanded, the culprit struck a theatrical pose and solemnly declared,

repeating Galileo's famous words: "It turns. 1 '

He was right, to some extent, after all, every inhabitant of the

globe "travels" not only as the earth rotates. He is transported with

still greater speed as the earth revolves around the sun. Every second this

planet of ours, with us and everything else on it, moves 30 km in space,

turning meanwhile on its axis. And thereby hangs a question not devoid

of interest: When do we move around the sun faster? In the daytime

or at night?

A bit of a puzzler, isn't it? After all, it's always day on one side of

the earth and night on the other. But don't dismiss my question as

senseless. Note that I'm asking you not when the earth itself moves

faster, but when we, who live on the earth, move faster in the heavens.

And that is another pair of shoes.

In the solar system we make two motions; we revolve around the

sun and simultaneously turn on the earth's axis. The two motions

add , but with different results, depending whether we are on the daylit

side or on the nightbound one.

Fig. 6 shows you that at midnight the speed of rotation is added to

that of the earth's translation, while at noon it is, on the contrary,

subtracted from the latter. Consequently, at midnight we move faster

in the solar system than at noon. Since any point on the equator travels

about half a kilometre a second, the difference there between midnight

and midday speeds comes to as much as a whole kilometre a second.

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Midday

Midnighi

Fig. 6. On the dark side we move around the sun faster

than on the sunlit side

Any of you who are good at geometry will easily reckon that for

Leningrad, which is on the 60th parallel, this difference is only half as

much. At 12 p.m. Leningraders travel in the solar system half a

kilometre more a second than they would do at 12 a.m.

THE CART-WHEEL RIDDLE

Attach a strip of coloured paper to the side of the rim of a cart-wheel

or bicycle tire, and watch to see what happens when the cart, or bicycle,

moves. If you are observant enough, you will see that near the ground

the strip of paper appears rather distinctly, while on top it flashes by

so rapidly that you can hardly spot it.

Doesn't it seem that the top of the wheel is moving faster than the

bottom? And when you look at the upper and lower spokes of the moving

wheel of a carriage, wouldn't you think the same? Indeed, the upper

spokes seem to merge into one solid body, whereas the lower spokes

can be made out quite distinctly.

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Incredibly enough, the top of the rolling-wheel does really move faster

than the bottom. And, though seemingly unbelievable, the explanation

is a pretty simple one. Every point on the rolling wheel makes two

motions simultaneously one about the axle and the other forward

together with the axle. It's the same as with the earth itself. The two

motions add, but with different results for the top and bottom of the

wheel. At the top the wheel's motion of rotation is added to its motion

of translation, since both are in the same direction. At the bot

torn rotation is made in the reverse direction and, consequently, must

be subtracted from translation. That is why the stationary observer

sees the top of the wheel moving faster than the bottom.

A simple experiment which can be done at convenience proves this

point. Drive a stick into the ground next to the wheel of a stationary

vehicle opposite the axle.Then take a piece of coal or chalk and make two

marks on the rim of the wheel at the very top and at the very bottom.

Your marks should be right opposite the stick. Now push the vehicle

a bit to the right (Fig. 7), so that the axle moves some 20 to 30 cm away

from the stick. Look to see how the marks have shifted. You will find

that the upper mark A has shifted much further away than the lower

one B which is almost where it was before.

Fig. 7. A comparison between the distances away from

the stick of points A and B on a rolling wheel (right) shows

that the wheel's upper segment moves faster than its lower

part

THE WHEEL'S SLOWEST PART

As we have seen, not all parts of a rolling cart-wheel move with the

same speed. Which part is slowest? That which touches the ground.

Strictly speaking, at the moment of contact, this part is absolutely

stationary. This refers only to a rolling wheel. For the one that spins

round a fixed axis, this is not so. In the case of a flywheel, for instance,

all its parts move with the same speed.

BRAIN-TEASER

Here is another, just as ticklish, problem. Could a train going from

Leningrad to Moscow have any points which, in relation to the railroad

track, would be moving in the opposite direction? It could, we find.

All the train wheels have such points every moment. They are at the

bottom of the protruding rim of the wheel (the bead). When the train

goes forward, these points move backward. The following experiment,

which you can easily do yourself, will show you how this happens.

Attach a match to a coin with some plasticine so that the match pro*

trades in the plane of the radius, as shown in Fig. 8. Set the coin together

with the match in a vertical position on the edge of a flat ruler and

hold it with your thumb at its point of contact C. Then roll it to and

fro. You will see that points F, E and D of the jutting part of the match

Fig. 8. When the coin is rolled

leftwards, points Ft E and

D of the jutting part of the

match move backwards

Fig. 9. When the train wheel

rolls leftwards the lower part

of its rim rolls the other way

/ig. 10. Top: the curve (a cycloid) described by every

point on the rim of a rolling cart-wheel. Bottom: the curve

described by every point on the rim of a train wheel

move not forwards but backwards. The further point D the end of the

match is from the edge of the coin, the more noticeable backward

motion is (point D shifts to D').

The points on the bead of the train wheel move similarly. So when

I tell you now that there are points in a train that move not forward

but backward, this should no longer surprise you. True, this backward

motion lasts only the negligible fraction of a second. Still there is,

despite all our habitual notions, a backward motion in a moving train.

Figs. 9 and 10 provide the explanation.

WHERE DID THE YACHT CAST OFF?

A rowboat is crossing a lake. Arrow a in Fig. 11 is its velocity vector.

A yacht is cutting across its course; arrow b is its velocity vector.

Where did the yacht cast off? You would naturally point at once to

point M. But you would get a different reply from the people in the

dinghy. Why?

They don't see the yacht moving at right angles to their own course,

because they don't realise that they are moving themselves. They think

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Fig. 11. The yacht is cutting across the rowboat's course. Arrows a and b designate

the velocities. What will the people in the dinghy see?

they're stationary, while everything around is moving with their own

speed but in the opposite direction. From their point of view the yacht

is moving not only in the direction of the arrow b but also in the direction

of the dotted line a opposite to their own direction (Fig. 12).

The two motions of the yacht the real one and the seeming one are

resolved according to the rule of the parallelogram. The result is that

the people in the rowboat think the yacht to be moving along the

diagonal of the parallelogram 06; that is also why they think the yacht

cast off not at point M, but at point /V, way in front of the rowboat

(Fig. 12).

Travelling together with the earth in its orbital path, we also plot

the position of the stars wrongly just as the people in the dinghy did

when asked where the yacht cast off from. We see the stars displaced

slightly forward in the direction of the earth's orbital motion. Of course,

the earth's speed is negligible compared with that of light (10,000

Fig. 12. The people in the dinghy think the yacht to be coming towards them

slantwise from point N

times less) and, consequently, this stellar displacement, known as

aberration of light, is insignificant. However, we can detect it with

the aid of astronomical instruments.

Did you like the yacht problem? Then answer another two questions

related to the same problem. Firstly, give the direction in which the

yachtsmen think the dinghy is moving. Secondly, say where the yachts*

men think the dinghy is heading. To answer, you must construct a parallelogram

of velocities on the vector a (Fig. 12), whose diagonal will

indicate that from the yachtsmen's point of view the dinghy seems to

be moving slantwise, as if heading for the shore.

speed and velocity ... Physics can be entertainment

SPEED AND VELOCITY. COMPOSITION

OF MOTIONS

HOW FAST DO WE MOVE?

A good athlete can run 1.5 km in about 3 min 50 pec the 1958

world record was 3 min 36.8 sec. Any ordinary person usually does,

when walking, about 1.5 metres a second. Reducing the athlete's rate

to a common denominator, we see that he covers seven metres every

second. These speeds are not absolutely comparable though. Walking,

you can keep on for hours on end at the rate of 5 km. p.h. But the

runner will keep up his speed for only a short while. On quick march,

infantry move at a speed which is but a third of the athlete's,

doing 2 m/sec, or 7 odd km. p.h. But they can cover a much greater

distance.

I daresay you would find it of interest to compare your normal walking

pace with the "speed" of the proverbially slow snail or tortoise.

The snail well lives up to its reputation, doing 1.5 mm/sec, or 5.4 metres

p.h. exactly one thousand times less than your rate. The other classically

slow animal, the tortoise, is not very much faster, doing usually

70 metres p.h.

Nimble compared to the snail and the tortoise, you would find yourself

greatly outraced when comparing your own motion with other

motions even not very fast ones that we see all around us. True,

you will easily outpace the current of most rivers in the plains and be

a pretty good second to a moderate wind. But you will successfully

vie with a fly, which does 5 m/sec, only if you don skis. You won't over-

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take a hare or a hunting dog even when riding a fast horse and you can

rival the eagle only aboard a plane.

Still the machines man has invented make him second to none for

speed. Some time ago a passenger hydrofoil ship, capable of 60-70 km.

p.h., was launched in the U.S.S.R. (Fig. 1). On land you can move faster

Fig. 1. Fast passenger hydrofoil ship

than on water by riding trains or motor cars which can do up to

200 km. p.h. and more (Fig. 2). Modern aircraft greatly exceed even

these speeds. Many Soviet air routes are serviced by the large TU-104

fig. 2. New Soviet ZIL-111 motor car

(Fig. 3) and TU-114 jet liners, which do about 800 km. p.h. It was

not so long ago that aircraft designers sought to overcome the "sound

barrier", to attain speeds faster than that of sound, which is 330 m/sec,

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or 1,200 km. p.b. Today this has been achieved. We have some small

but very fast supersonic jet aircraft that can do as much as 2,000

km.p.h.

There are man-made vehicles that can work up still greater speeds.

The initial launching speed of the first Soviet sputnik was about

fig. 3. TU-104 jet airliner

8 km/sec. Later Soviet space rockets exceeded the so-called

velocity, which is 11.2 km/sec at ground level.

The following table gives some interesting speed data.

escape

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RACING AGAINST TIME

Could one leave Vladivostok by air at 8 a.m. and land in Moscow

at 8 a.m. on the same day?

I'm not talking through my hat. We can really do that. The answer

lies in the 9-hour difference in Vladivostok and Moscow zonal times.

If our plane covers the distance between the two cities in these 9 hours,

it will land in Moscow at the very same time at which it took off from

Vladivostok. Considering that the distance is roughly 9,000 kilometres,

we must fly at a speed of 9,000:9=1,000 km. p.h., which is quite

possible today.

To "outrace the Sun" (or rather the earth) in Arctic latitudes,

one can go much more slowly. Above Novaya Zemlya, on the 77th parallel,

a plane doing about 450 km. p.h. would cover as much as a definite

point on the surface of the globe would cover in an identical space of

time in the process of the earth's axial rotation. If you were flying in

such a plane you would see the sun suspended in immobility. It would

never set, provided, of course, that your plane was moving in the

proper direction.

It is still easier to "outrace the Moon" in its revolution around the

earth. It takes the moon 29 times longer to spin round the earth than

it takes the earth to complete one rotation (we are comparing, naturally,

the so-called "angular", and not linear, velocities). So any ordinary

steamer making 15-18 knots could "outrace the Moon" oven in the

moderate latitudes.

Mark Twain mentions this in his Innocents Abroad. When sailing

across the Atlantic, from New York to the Azores "... wo had balmy

summer weather, and nights that were even finer than the days. We had

the phenomenon of a full moon located just in the same spot in the

heavens at the same hour every night. The reason for this singular conduct

on the part of the moon did not occur to us at first, but it did afterward

when we reflected that we were gaming about twenty minutes every day,

because we were going east so fast we gained just enough every day

to keep along with the moon. "

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THE THOUSANDTH OF A SECOND

For us humans, the thousandth of a second is nothing from the angle

of time. Time intervals of this order have only started to crop up in

some of our practical work. When people used to reckon the time according

to the sun's position in the sky, or to the length of a shadow

(Fig. 4), they paid no heed to minutes, considering them even unworthy

Fig. 4. How to reckon the time "according to the

position of the sun (left), and by the length of a shadow

(right)

of measurement. The tenor of life in ancient times was so unhurried

that the timepieces of the day the sun-dials, sand-glasses and the

like had no special divisions for minutes (Fig. 5). The minute hand

first appeared only in the early 18th century, while the second sweep

came into use a mere 150 years ago.

But back to our thousandth of a second. What do you think could

happen in this space of time? Very much, indeed I True, an ordinary

train would cover only some 3 cm. But sound would already fly 33 cm

and a plane half a metre. In its orbital movement around the sun, the

earth would travel 30 metres. Light would cover the great distance of

300 km. The minute organisms around us wouldn't think the thousandth

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of a second so negligible an amount of time if they could think of

course. For insects it is quite a tangible interval. In the space of a

second a mosquito flaps its wings 500 to 600 times. Consequently in

the space of a thousandth of a second, it would manage either to raise

its wings or lower them*

We can't move our limbs as fast as insects. The fastest thing we can

do is to blink our eyelids. This takes place so quickly that we fail even

to notice the transient obscurement of our field of vision. Few know,

though, that this movement, "in the twinkling of an eye" which has

Fig. 6. An ancient water clock (loft) and an old pocketwatch

(right). Note that neither has the minute

hand

become synonymous for incredible rapidity is quite slow if measured

in thousandths of a second. A full "twinkling of an eye" averages as

exact measurement has disclosed two-fifths of a second, which gives

us 400 thousandths of a second. This process can be divided into the

following stages: firstly, the dropping of the eyelid which takes 75-90

thousandths of a second; secondly, the closed eyelid in a state of rest,

which takes up 130-170 thousandths; and, thirdly, the raising of the

eyelid, which takes about 170 thousandths.

As you see, this one "twinkling of an eye" is quite a considerable time

interval, during which the eyelid even manages to take a rest. If we

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could photograph mentally impressions lasting the thousandth of a

second, we would catch in the u

twinkling of an eye'* two smooth motions

of the eyelid, separated by a period during which the eyelid would

be at rest.

Generally speaking, the ability to do such a thing would completely

transform the picture we get of the world around us and we would see

the odd and curious things that H. G. Wells described in his New Accelerator.

This story relates of a man who drank a queer mixture which

caused him to see rapid motions as a series of separate static phenomena.

Here are a few extracts.

"'Have you ever seen a curtain before a window fixed in that way

before?'

"I followed his eyes, and there was the end of the curtain, frozen, as

it were, corner high, in the act of flapping briskly in the breeze.

"'No,

1 said I, 'that's odd.'

"'And here,' he said, and opened the hand that held the glass. Naturally

I winced, expecting the glass to smash. But so far from smashing

it did not even seem to stir; it hung in mid-air motionless. 'Roughly

speaking,' said Gibberne, 'an object in these latitudes falls 16 feet in

a second. This glass is falling 16 feet in a second now. Only you see,

it hasn't been falling yet for the hundredth part of a second. [Note also

that in the first hundredth of the first second of its downward flight a

body, the glass in this case, covers not the hundredth part of the distance,

but the 10,000th part (according to the formula S=U2 gt*). This

is only 0.5 mm and in the first thousandth of the second it would be

only 0.01 mm.l

"'That gives you some idea of the pace of my Accelerator.' And he

waved his hand round and round, over and under the slowly sinking

glass.

"Finally he took it by the bottom, pulled it down and placed it

very carefully on the table. 'Eh?' he said to me, and laughed....

"I looked out of the window. An immovable cyclist, head down and

with a frozen puff of dust behind his driving-wheel, scorched to overtake

a galloping char-a-banc that did not stir....

"We went out by his gate into the road, nnd there we made a minute

examination of the statuesque passing traffic. The top of the wheels

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and some of the legs of the horses of this char-a-banc, the end of the

whip lash and the lower jaw of the conductor who was just beginning

to yawn were perceptibly in motion, but all the rest of the lumbering

conveyance seemed still. And quite noiseless except for a faint rattling

that came from one man's throat! And as parts of this frozen

edifice there were a driver, you know, and a conductor, and eleven

people!...

"A purple-faced little gentleman was frozen in the midst of a violent

struggle to refold his newspaper against the wind; there were many evidences

that all these people in their sluggish way were exposed to a

considerable breeze, a breeze that had no existence so far as our sensations

went....

"All that I had said, and thought, and done since the stuff had begun

to work in my veins had happened, so far as those people, so far as the

world in general went, in the twinkling of an eye...."

Would you like to know the shortest stretch of time that scientists

can measure today? Whereas at the beginning of this century it was

only the 10,000th of a second, today the physicist can measure the

100,000 millionth of a second; this is about as many times less than a

second as a second is less than 3,000 years!

THE SLOW-MOTION CAMERA

When H. G. Wells was writing his story, scarcely could he have

ever thought he would see anything of the like. However he did live

to see the pictures he had once imagined, thanks to what has been

called the slow-motion camera. Instead of 24 shots a second as ordinary

motion-picture cameras do this camera makes many times more.

When a film shot in this way is projected onto the screen with the

usual speed- of 24 frames a second, you see things taking place much

more slowly than normally high jumps, for instance, seem unusually

smooth. The more complex types of slow-motion cameras will almost

Simula H. G. WeiIs 's world of fantasy.

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