Class 7 Science Chapter 8: Measurement of Time and Motion

A Question Worth Asking

Prerna watches sprinters on television, timed so finely that a camera can separate two runners who seem to cross the line together. Her school has a stopwatch, her mother a wristwatch, her uncle a Braille watch and a talking one. Then she wonders about people long ago.

How Do You Measure Time With Nothing?

That is the real question. If you have no watch, no clock and no battery, what is left? The answer people found is a good piece of thinking, and it starts with noticing rather than building.

Noticing First

People saw that a great many things in nature happen again and again, with a definite gap between each time. The Sun rises and sets. The Moon goes through its phases. The seasons come round. None of that needed inventing; it only needed watching.

Why That Is Enough

Here is the useful step. If an event repeats with the same gap every time, you can count the repeats instead of measuring anything at all. Three sunsets since a thing happened is a real statement about how long ago it was.

So Calendars Came First

That is why the first timekeeping was a calendar rather than a clock. A day was defined by one cycle of the Sun rising and setting, and longer stretches were counted in days. It works, and it needs nothing but attention.

But Within a Day?

A calendar cannot tell you when to meet somebody this afternoon. So the next quest was for something that could measure the smaller intervals inside a single day, and for that people did have to build things.

Four Devices

They built several: sundials, water clocks, hourglasses and candle clocks. It is worth seeing what they have in common, because the idea behind all four is the same one as the calendar.

The Sundial

Stand an object in sunlight and it throws a shadow. As the day goes on, that shadow swings round and changes its length, dependably and in the same way every day. Read where the shadow has reached and you have the time.

The Water Clock, First Kind

Water trickles at a steady rate, which is the useful property. In one sort of water clock, water flows out of a vessel that has markings cut into it for time. As the level falls past each mark, you know how long has gone.

The Water Clock, Second Kind

The other sort is cleverer and worth picturing. A bowl with a fine hole in its bottom is set floating on water. Water seeps in through the hole, the bowl gradually fills, and after a fixed time it sinks. Then somebody lifts it out and floats it again.

The Hourglass

Sand behaves like water for this purpose. In an hourglass, sand runs from one bulb through a narrow neck into the other, and the time is measured by how much has gone across. Turn it over and it is ready again.

The Candle Clock

A candle burns down at a fairly steady rate, so a candle with markings along it shows the passage of time as it burns past each one. This is the only one of the four that destroys itself in the process of working.

The Pattern Behind All Four

Look at what every one of them does. Each finds something that changes at a dependable rate, shadow, water, sand or wax, and then reads off how far that change has got. That is all a clock has ever been.

Build One Yourself

You can make the first sort of water clock from a plastic bottle. Cut it in half, make a small hole in the cap with a drawing pin, and stand the top half upside down inside the bottom half. Fill the top with water, coloured with a drop of ink so the level shows.

Marking It

The water drips through into the lower half. Using a watch, mark the water level every minute until it has all gone through. Those marks are your scale, and now your clock is finished, with a watch used only to make it rather than to run it.

The Biggest One Ever Built

Some of these devices were enormous. The Samrat Yantra at the Jantar Mantar in Jaipur, put up about three hundred years ago, is the largest stone sundial in the world. Timekeeping on that scale was serious science, not decoration.

And an Indian Unit of Time

The floating bowl of the ghatika-yantra had its hole cut so precisely that it took twenty-four minutes to fill and sink. That interval became a unit of time in its own right, called a ghatika or ghati.

Sixty to a Day

The arithmetic is tidy. A day of twenty-four hours divides into sixty equal ghatis of twenty-four minutes each. It became the standard measure of time in India and stayed in use until the end of the nineteenth century.

Scenario:

A village school loses its electricity for a whole day, and with it the wall clock. The teacher needs to run four forty-minute lessons and a short break, and nobody has a working watch. One student suggests watching the shadow of the flagpole. Another says they should use the school water cooler by letting it drip into a bucket. A third says the first idea is better than the second as it stands, but that the second could be made to work with twenty minutes of preparation and something the first can supply.

Analysis:

The third student has seen the real point, which is that a clock needs a SCALE as well as a steady change. The flagpole idea works at once because the Sun provides both: the shadow swings round dependably, and anybody who has watched it before roughly knows where it falls at a given hour. That is a sundial. The dripping cooler has the steady change too, since water trickles at a dependable rate, but it has no markings, so a bucket filling up tells nobody anything. What it needs is exactly the preparation step from the bottle activity: use a known clock to mark the level at intervals, and after that the device reads its own time. The thing the first idea can supply is that known clock. Watch the flagpole shadow, mark the bucket every time the shadow moves a set amount, and the water clock is calibrated. Note that neither device creates time. Each holds a steady change, and the marks are what turn it into a measurement.

Activity:

Find something in your home that changes at a steady rate, such as a dripping tap, a melting ice cube or a shadow on a wall. Watch it for ten minutes with a clock and make marks. Then write down what interval your home-made clock can measure, and one interval it is useless for.

Title:

Try it at home: the bottle water clock

Safety:

An adult must do the cutting. Mind the point of the drawing pin, and work over a tray or a sink.

What You Notice:

Pour the water back into the top and watch it drip again. Each time the level reaches one of your marks, another minute has passed. The watch made the scale; the bottle keeps the time.

The Samrat Yantra at Jantar Mantar in Jaipur is the largest stone sundial in the world.

The hole in a ghatika-yantra bowl was cut so that it took exactly 24 minutes to fill and sink.

A ghati stayed India standard unit of time until the end of the nineteenth century.

A clock is anything that repeats dependably. Shadow, water, sand, wax.

Sixty ghatis to a day, twenty-four minutes each.

A Clock You Can Build From a Stone

The last topic needed a steady change to read. A pendulum offers one, and it offers something better than a trickle of water: it repeats. That repetition is what eventually put pendulums inside clocks.

What One Is

A simple pendulum is about as simple as the name suggests. A small metallic ball, which is called the bob, hangs by a long thread from a rigid support. Rigid matters. If the support wobbles, nothing below it can be trusted.

Three Positions to Name

Leave the bob hanging and it comes to rest. That resting place is called the mean position. Now move the bob slightly to one side and release it. It swings across, reaching a furthest point on each side, and those two are the extreme positions.

Oscillatory, and Periodic

The swinging to and fro is called oscillatory motion. It is also described as periodic, and the word means something precise: the motion comes round again after a fixed interval of time. Not roughly, and not eventually. After a fixed interval.

Counting One Swing

Now the part most people get wrong, so take it slowly. Starting from the mean position O, the bob moves to one extreme A, turns, crosses to the other extreme B, turns again, and comes back to O. That whole journey is one oscillation.

The Same Thing, Counted Differently

You can begin the count anywhere, as long as you end where you started. Beginning at extreme A, the bob goes across to extreme B and comes back to A, and that is also one oscillation. Both descriptions cover exactly the same amount of swinging.

Why That Is Worth Being Careful About

It is easy to count a half swing as a whole one. A bob going from A to B has done half an oscillation, not one, and counting that way would make every time period twice too small. The test is whether the bob has returned to where you began counting.

The Time Period

The time the pendulum needs for one whole oscillation is called its time period. Because the motion is periodic, that figure is the same swing after swing, which is precisely why it can be used for timekeeping.

Setting One Up

To measure it, collect a string about 150 centimetres long, a heavy metal ball with a hook or else a stone for the bob, a stopwatch and a ruler. Tie the bob to one end, and fix the other end to a rigid support so that about 100 centimetres hangs between support and bob.

Starting It Properly

Let the bob come to rest first, then hold it gently, move it slightly to one side and let go. Two cautions. Do not push it as you release it, and keep the string taut. A pushed bob or a slack string spoils the measurement before it starts.

Why Time Ten, Not One?

Here is the useful trick. Measure the time for ten oscillations, not for one, and then divide that time by ten. Repeat the whole thing three or four times.

And the Reason It Helps

One swing takes a second or two, and a human hand on a stopwatch is late by a noticeable fraction of that. Spread the same mistake across ten swings and dividing by ten shrinks it to a tenth. The pendulum has not become more accurate; your measurement has.

Now the Unit Itself

All of that timing needs a unit, and the second is the SI unit of time. Its symbol is s. The larger units are the minute, with symbol min, and the hour, with symbol h. Sixty seconds make one minute, and sixty minutes make one hour.

How to Write Them

The rules for writing these are strict and worth learning once. The unit names take a lowercase first letter, except at the start of a sentence. The symbols s, min and h are lowercase too, and always singular, so ten seconds is 10 s.

Two More Rules

No full stop is written after a symbol, unless it happens to fall at the end of a sentence. And always put a space between the figure and the unit. So 10 s is right and 10s is not.

And Two Common Mistakes

Finally, two things that look reasonable and are simply wrong. Writing sec for second is incorrect, and so is writing hrs for hour. The symbols are s and h, and they do not take an s for the plural.

Why All This Precision Matters

It can seem fussy until you see where fine timing is used. In sport, timing devices record to a hundredth or even a thousandth of a second, which is a millisecond, and that is how two sprinters crossing together can be separated.

Beyond Sport

In medicine, an ECG machine measures millisecond variations in heartbeats to find problems. Digital recordings capture sound thousands of times a second. And phones and computers handle signals in microseconds, each a millionth of a second.

Scenario:

Two groups set up pendulums with 100 centimetres of string. Group A times one swing with a stopwatch, gets 2.3 seconds, and reports a time period of 2.3 s. Group B times ten swings, gets 20 seconds, divides by ten and reports 2.0 s. Group A then repeats its single-swing measurement four times and gets 2.3, 1.7, 2.5 and 1.9 seconds, which it averages to 2.1 s. The teacher says Group B method is better even though both ended up near the same number, and that Group A spread of readings is the evidence.

Analysis:

The teacher is right, and the spread is the whole argument. Both groups have the same pendulum, so the true time period is one number. Group A readings range from 1.7 to 2.5 seconds, which is a spread of 0.8 seconds on a quantity of about two. That spread is not the pendulum changing its mind; it is the human hand on the stopwatch, late by a different fraction of a second on each attempt. Group B has exactly the same hand and the same delay, but spread across ten swings, so dividing by ten shrinks the error to a tenth of what it was. That is why their single reading is already better than Group A average of four. Note what averaging can and cannot do. Group A average did help, and it landed close to the right answer, but it needed four runs to get there and it still carries the larger uncertainty. The better method is the one that makes the error small to begin with.

Activity:

Set up a pendulum with a thread and any small heavy object, and time one swing five times over. Write down all five readings and the spread between the largest and smallest. Then time ten swings once and divide by ten, and compare that answer with your five.

Title:

Try it at home: your own pendulum

Safety:

Keep the pendulum away from faces and from anything breakable, and do not swing it hard.

What You Notice:

Your four time periods should agree closely, because the motion is periodic. If one reading is far out, it is almost certainly the timing rather than the pendulum.

A pendulum is periodic, which simply means it comes round again after a fixed interval.

Writing sec for second or hrs for hour is incorrect: the symbols are s and h.

An ECG machine reads variations in heartbeats measured in thousandths of a second.

Middle, out, across, back to the middle. That is one oscillation.

Time ten and divide by ten; one swing alone is too short to time well.

Who Is Faster?

Picture a hundred metre race on a straight track. Everybody starts from the same line together, and after a few seconds they are no longer together. Which of the two is the faster runner? The answer is so obvious that it is worth saying slowly.

The Obvious Answer, Stated Carefully

Whoever is ahead of the others at a given instant is running faster than them. Put the other way round, whoever got further in the same length of time is faster. Distance, compared over a shared interval of time. That is the whole of it.

Which Gives Us a Quantity

So what each object gets through in one agreed stretch of time decides which is faster and which slower. Once that is clear, the idea can be turned into a number rather than a comparison, and the number is called speed.

What Speed Is

Compare the distances two objects move in unit time and you can say which is faster. So we call the distance an object covers in unit time its speed. The unit of time may be one second, one minute or one hour, whichever suits.

How to Calculate It

To find the speed of something you need two things: the total distance it covered, and the time it took to cover that distance. Then the speed is the first divided by the second.

The Formula

Written out, speed equals the total distance covered, divided through by the total time taken. It is worth writing it with both words total in place, because they matter later when we come to average speed.

What Unit Does That Give?

The unit follows from the formula rather than being chosen. Speed is a distance divided by a time, so its unit must be a length unit divided by a time unit. In SI that means metre divided by second, which is written m/s.

And a More Everyday Unit

Other units work the same way. Measure the distance in kilometres and the time in hours and the speed comes out in kilometre per hour, written km/h, which is what a vehicle speedometer displays.

A Worked Example

Swati school is 3.6 kilometres from her house, and she cycles there in 15 minutes. What is her speed in metres per second? The formula wants a distance over a time, so first put both into SI units.

Doing the Conversion

3.6 kilometres is 3.6 times 1000, which is 3600 metres. Fifteen minutes is 15 times 60, which is 900 seconds. Now divide: 3600 metres divided by 900 seconds gives 4 metres per second. So her speed is 4 m/s.

Why the Conversion Came First

Notice the order of work there. The units were converted before the division, not after. Dividing 3.6 by 15 would give 0.24, which is a perfectly real number in kilometres per minute and no use at all if metres per second was what was wanted.

Now Turn the Relation Round

Here is where one formula becomes three. If you already know the speed and the time, you can get the distance by multiplying instead of dividing: total distance covered equals speed times total time taken.

An Example of That

Raghav rides to the next city on a bus doing 50 kilometres per hour, and it takes him 2 hours. How far is the city? Distance equals speed times time, so 50 kilometres per hour times 2 hours gives 100 kilometres.

And the Third Arrangement

If instead you know the distance and the speed, you can find the time: total time taken equals total distance covered divided by speed. The relation has not changed at all; only which quantity is missing has changed.

An Example of That Too

A train travels at 90 kilometres per hour. How long will it take to cover 360 kilometres? Time equals distance over speed, so 360 kilometres divided by 90 kilometres per hour gives 4 hours.

One Relation, Three Questions

So there is really only one relation here, and the three forms simply answer three different questions. Know any two of speed, distance and time and the third follows. That is why this is worth learning once rather than three times.

A Caution About All of It

In every example above, the speed was found by dividing a total distance by a total time. But the object might not have travelled at the same speed all the way. It may have gone slower at times and faster at others.

So These Are Average Speeds

The figure that comes out of the division is therefore an average speed. Swati bicycle was not doing exactly 4 m/s at every moment of her journey; 4 m/s is what her whole trip works out to. In this book, the word speed is used to mean average speed.

Scenario:

A school hires a bus for a trip of 150 kilometres. The company says the bus averages 50 kilometres per hour, so the teacher allows three hours. On the day, the bus reaches halfway in one hour, which is faster than promised, so the students conclude they will arrive early. They then crawl through a town for forty minutes, and arrive almost exactly three hours after setting off. One student says the company figure was wrong twice over, since the bus was never actually doing 50 km/h. Another says the figure was right all along and the students misunderstood what it meant.

Analysis:

The second student is right, and the misunderstanding is exactly the one this topic warns about. The company quoted an AVERAGE speed. That is what you get by dividing a total distance by a total time. Check it. 150 kilometres divided by 50 kilometres per hour gives 3 hours. That is what the journey actually took. So the figure predicted the arrival correctly. What it never claimed was that the bus would be doing 50 km/h at any given moment. In fact it almost certainly never was. It covered the first 75 kilometres in an hour, an average of 75 km/h over that stretch. Then it crawled through a town well below 50. The first student mistake is to treat an average as a description of every moment. The students error was the same one in the other direction. They took one fast stretch as evidence about the whole trip. An average can only be compared with an average.

Activity:

Find the distance from your home to your school, and time the journey three times. Work out the average speed for each trip, and write down how much the three answers differ. Then describe one part of the route where you know you were going faster than your average, and one where you were slower.

Title:

Try it yourself: measure your own walking speed

Safety:

Choose a clear, flat space with nothing to trip over, and do not run indoors.

What You Notice:

Most people walk at somewhere between 1 and 2 m/s. Notice that each answer is an average for that whole 20 metres, even though you were certainly not at one exact speed the whole way.

A speed of 72 km/h is the same as 20 m/s, since 1000 metres go into a kilometre and 3600 seconds into an hour.

Every speed in this book is really an average speed, because objects seldom hold one speed for long.

Vehicles carry an instrument that measures and displays their speed in km/h.

Cover the one you want: speed under distance, time beside it.

More ground in the same time means faster. That is the whole comparison.

Back to a Straight Line

You met linear motion in Grade 6, and it is worth restating simply. When a body travels down a straight line, we call its motion linear motion. That is the whole definition, and notice what it does not mention.

What It Does Not Say

It says nothing about speed. An object in linear motion may be crawling, racing, speeding up or slowing down. All that is being claimed is that the path is straight. Keeping that separate from speed is the key to this topic.

A Train Between Two Stations

Take a train on a track that runs straight between two adjacent stations. Its motion along that stretch is linear motion. Now follow it from one station to the next and watch what the speed does, because that is where the interesting part is.

The Four Stages

It starts from the first station, A, at a slow speed. Then it moves at a faster speed. Then it slows down. Then it pulls up at the next station, D. Anybody who has sat on a train recognises all four of those.

But Something Happens in the Middle

For part of the way, between B and C, the train holds one constant speed. Constant simply means unchanging. For that stretch, and only that stretch, the train is doing the tidiest thing it does all journey.

Naming the Tidy Case

That stretch has a name. Motion in a straight line in which the object covers equal distances in each equal interval of time is called uniform linear motion. Both halves of that matter: the straight path, and the unchanging speed.

The Test You Can Actually Apply

The definition is also a method. Take readings at equal intervals of time, say every ten minutes, and look at the distances covered in each interval. If they are all the same, the motion is uniform. If they differ, it is not.

Two Trains, One Table

Suppose you have the positions of two trains, X and Y, recorded every ten minutes from ten o clock to eleven. Train X covers equal distances in those equal intervals, so Train X is in uniform linear motion. Train Y does not, so Train Y shows non-uniform linear motion.

Notice What Decided It

The deciding evidence was a table of numbers, not a feeling about which train seemed smoother. That is worth holding on to, because uniform motion is a claim you can check, and checking it means comparing distances across equal slices of time.

Both Are Still Linear

And notice that Train Y is still in linear motion. It is travelling along a straight track. What it is not is uniform. The path and the speed are two separate questions, and the names keep them separate: linear describes the path, uniform describes the speed.

The Honest Admission

Now the part of this topic that is easy to skip and worth more than the rest. Uniform linear motion is an idealisation. It is a clean case, useful for thinking with, and in everyday life we seldom find it.

Why We Seldom Find It

Objects rarely move at a constant speed across long distances or over long stretches of time. Roads bend, traffic slows, trains stop at stations, and a runner tires. The train in our example managed it only between B and C, and not for the whole journey from A to D.

Which Explains the Previous Topic

And that is exactly why average speeds exist. If objects really did hold one speed, a single speed would describe a whole journey perfectly and no average would be needed. Because they do not, dividing a total distance by a total time is the honest thing to do.

So the Two Topics Fit Together

Put them side by side. Average speed is not a rough substitute for something better; it is the right tool for a world where uniform motion hardly happens. The idealisation tells you why the average is necessary.

What a Vehicle Can Tell You

There is one way to watch this directly. Scooters, motorbikes, cars and buses carry an instrument that measures and displays the vehicle speed in kilometres per hour. Watch it on any journey and the needle rarely sits still.

What the Needle Shows

That moving needle is non-uniform motion, seen as it happens. On a long clear stretch it may hold steady for a while, and for those few minutes the vehicle is in uniform motion. Then a turning or a traffic light arrives, and it is not.

Scenario:

A school bus covers the 30 kilometres to school along a straight highway. The driver says he keeps a steady 40 kilometres per hour the whole way. A student at the front writes down the odometer reading every ten minutes. For the five intervals she gets 5 km, 8 km, 7 km, 6 km and 4 km. The driver says those figures prove his point. They add up to 30 kilometres in fifty minutes. That is about 36 km/h, close enough to what he claimed. The student says they prove the opposite. She says the driver has answered a different question.

Analysis:

The student is right, and the two of them are using the word speed for two different things. The driver claim was that he holds a steady speed. That is a claim about uniform motion. The test for it is whether equal distances are covered in equal intervals of time. The student readings are taken at equal ten-minute intervals, which is what makes them usable. The distances are 5, 8, 7, 6 and 4 kilometres. Those are not equal. So the motion is non-uniform, and the driver own journey refutes his claim. What he has calculated instead is an average speed, by dividing a total distance by a total time. That figure is perfectly correct. It tells you nothing about steadiness. Note that the motion is still LINEAR throughout, since the highway is straight. Linear describes the path and uniform describes the speed, and the bus satisfies the first without satisfying the second.

Activity:

On your next journey of any length, write down the time whenever you pass a landmark you can identify, and estimate the distance between landmarks. Work out whether the distances covered in roughly equal stretches of time were equal, and then write one sentence saying whether the motion was uniform.

Title:

Try it yourself: uniform or not?

Safety:

Use a clear path with nothing to trip on, and walk rather than run.

What You Notice:

In the even walk the interval times come out nearly equal, which is close to uniform linear motion. In the second they fall steadily, which is non-uniform. Notice how hard the first one is to do exactly, which is what makes uniform motion an idealisation.

Uniform linear motion is an idealisation, which is why average speeds are needed at all.

A train between two stations is non-uniform overall, even though part of the trip is uniform.

The instrument in a vehicle that measures and displays its speed reads in km/h.

Straight line plus unchanging speed equals uniform. Drop either and it is not.

Equal times, equal distances. That is the test, and it fits on one line.