You can feel and measure accelerations directly

You were born on a planet where acceleration is omnipresent – both in the natural and in the made world. Release a stone held out in your hand and it accelerates. The same applies to amusement park rides where the ride container often falls in a controlled way. Watch a jet take off and it too accelerates, as it does when it lands (but now the acceleration is negative – more on that later). However, in all these cases you're noticing the acceleration from a detached point of view. How about getting closer to the action?

Catch a wave and you feel the lurch of acceleration as you take off on your surfboard. Sit in the jet and you also feel the lurch, as you do if you're in the fairground ride when it's released. You can feel these lurches – no need to keep your eyes open. That's because we've evolved on a planet where there is a gravitational field, and so we've developed acceleration sensors. These are built into the inner ear, and they act primarily as orientation sensors, so you can figure out which way is up. The gravitational field, acting on the masses in these sensors, along with every other mass in the vicinity, deflects them. Change the orientation and you change the direction of deflection. But you can also deflect them by accelerating your head. The origin of motion sickness is a mismatch between visual and accelerometer inputs. Sit in deep in the hull of a ship and your eyes tell you that there is no acceleration – but your inner ears are sending exactly the opposite message.

Someone on the shoreline will also expect you to be experiencing acceleration, as will another sailor on a vessel sailing a parallel course. Perhaps the acceleration is something that Alice, Bob and Charlie, from their three different points of view, will always agree about. It's not likely that they'll agree about your velocity – but more on that later.

How to notice an acceleration

So you can notice an acceleration, and the lurches of acceleration we've introduced. Examples we've picked so far might suggest the following effects of an acceleration:

• You're noticed to slow down.
• You're noticed to speed up.
• You're noticed to be moving in a new direction.

Take a moment to match these to the following examples:

1. A plane taking off.
2. A plane landing.
3. A braking car.
4. A car pulling away from traffic lights.
5. A car cornering.
6. A kayak pitching in a rough sea.

You might also find it useful to match the three categories (speeding up, slowing down and changing direction) to examples of your own.

You can draw together all of these seemingly disparate statements into a single statement:

acceleration changes velocity.

That's quite some claim, and a significant synthesis, so we'll spend some time exploring the idea.

Thinking with units

We hope you managed to do the matching well, and to add your own examples to our list:

• 2, 3
• 1, 4
• 5, 6

We've already met acceleration, in episode 1, as a consequence of force acting on mass. There it had the units of (metre/second) every second. Lots of physical quantities have units like this, and so can be thought of as instructions to accumulate. Here are two more: Energy, power and time: Δ energy = power × time interval (expressed using units, this is joule = joule second^-1 × second). Number, activity and time: Δ number = activity × time interval (expressed using units, this is no units = second^-1 × second). (Remember: Δ x is just shorthand for change in x.)

Acceleration and velocity follow this pattern: Δ velocity = acceleration × time interval (expressed using units, this is metre second^-1 = metre second^-2 × second. In this topic this pattern of accumulation is particularly important. This is the first of many meetings.

We'll get to look at velocity in more detail in this episode. For now, it's simply the result of accumulating acceleration.

Understanding acceleration as an accumulation

Acceleration is all about driving an accumulation.

For every second that the acceleration occurs, the velocity changes by a set amount. This change could be positive or negative, so the acceleration could be positive or negative.

A constant positive acceleration adds a fixed number of metres/second to the velocity in each second.

A constant negative acceleration subtracts a fixed number of metres/second from the velocity in each second.

Usually you'll stick to one-dimensional motion. The magnitude of the increment or decrement is fixed by the magnitude of the acceleration. Whether it's an increment or decrement is fixed by the direction, shown by a sign in the simple one-dimensional case.

However, do remember that acceleration is a vector, with all that this implies about changing velocity in two and three dimensional cases.

It's not easy to measure velocity

Just how fast am I going? Seems an easy question, but appearances are very definitely deceptive.

Here's a snippet of conversation between a young child (about six years old) and his dad, both passengers on a train.

Boy: When will it go fast?

Dad: It is going fast.

Boy: It's not going fast.

Dad: You might not feel it's going fast. Don't you remember being in an aeroplane? That didn't feel fast either.

The boy mumbles something inaudible and looks unimpressed with this line of argument.

But Dad's position is a difficult one. There are no reliable guides to sensations of velocity. We simply do not have a built-in velocity meter, comparable to our built-in acceleration meter.

And this lack of a built-in meter is not really accidental as there are good reasons to believe such a thing could not be engineered. Velocity all depends on a point of view, as we'll see shortly. For now it's enough to notice that the boy and his dad were not moving apart, so from one point of view (the child's?), their velocity was zero. However, they were definitely getting closer and closer to London, so from another point of view their velocity could not be zero.

So another line of attack with the child's initial question, likely to have left him even less impressed, would be:

Dad: Well, it all depends on your point of view.

And so it does – always. Simple questions about how fast one is going may not have simple, unique answers. One has to ask a less simple question to get a more simple answer. An essential additional component of the enquiry is to make explicit the point of view. To miss that out, like this:

Innocent: How fast am I going?

makes lots of hidden assumptions. Only a more comprehensive question can better define the answer space.

Learning: How fast am I moving away from that fencepost?

or

Egocentric: How fast is that fencepost moving away from me?

And later you'll see that a more sophisticated and complete approach still might begin here:

Sophisticate: By how much is the distance between me and the fencepost increasing each second?

Measurements of How fast? don't always depend on first measuring distance and time

But we get ahead of ourselves. Here, just remember that you need to agree on a point of view before you can settle a How fast? question. Often the point of view is implicit: this may or may not lead to confusion – it depends on whether others share your implicit assumptions.

There is one curiosity about measurements of velocity that's worth mentioning here – we can do it remotely. In some cases, very remotely. This relies on the Doppler effect. This is worth dwelling on because you don't have first to measure distance and then time. Instead you simply count.

Bats and policemen are both interested in velocities of approach and recession, and both use similar physics, although very different technologies.

Speed enforcement officer: What's the velocity of that car?

Bat: What's the velocity of that insect?

Or, even:

Physics bat: What's the velocity of that tree?

Bats do this by engaging in active sensing – firing out a burst of vibrations and then measuring the shift in frequency of the reflections.

Doctors use a similar active sensing technique for measuring blood-flow, but planetary scientists use a variation. They're obliged to, because the trip time is huge: the interval between the emitted and reflected beams could be years.

Full details are in the SPT: Radiations and radiating topic.

Starting a train journey

It's time to leave. You've found a window seat, farthest from the platform, and are settling in for the long train journey down to London. Still busy with getting your laptop out, you feel some gentle movement and hear a roar and then you're off – but wait a second – backwards!

After a moment's panic you realise that it's the local train on the adjacent platform that's moving off in the same direction as you, but a little faster. Look out of the other window, and you see the platform sliding away behind you, moving away from you in the expected direction.

The panic subsides.

The couple behind you had a moment as well:

Alice: That's weird, I thought we were off to Hereford.

So much about motion depends on our assumed point of view, often not questioned until we have a moment. This is often implicit, remaining unstated. As you might imagine, that can lead to difficulties.

• From our train, Alice notices the local train drawing away from us.
• Alice also notices the platform going backwards.
• A passenger, Bob, in the local train notices our train moving backwards.
• A waiting passenger on our platform, Charlie, sees both trains moving off in the same direction, but at different velocities.

So it seems that Alice, Bob and Charlie all notice different velocities for the:

• Platform.
• The local train.
• The London train.

Many velocities – depending on your point of view

Here are three separate points of view. So who is correct about the velocities of these trains and platforms?

Everyone and no-one. All three need to be rather more careful and rigorous as they seek to turn their implicit noticing into unambiguous physical statements that everyone can agree on.

You need to be careful before you can agree on measured quantities.

Understanding motion

Being at rest is just a rather special case of moving alongside something, so that its separation from you does not change with time. There's no privileged point of view in the universe, where the most important person sits, and rules on what is and what is not at rest. Instead you have to be careful to state your point of view. Only once that's done is it worth reporting on the velocities that you measure.

This shouldn't be that surprising: velocity determines changes in position in the same way that acceleration determines changes in velocity. And to determine the position of something you need to say where it is with respect to some chosen location (or point of view).

Several different choices of points of view will result in several different velocities – unless, of course, the chosen points of view are all moving along together and report each other as being at rest.

All motion depends on a point of view. These are all too often chosen without careful thought, as being obvious. Alice, sitting on a moving train will naturally describe the seats around her as stationary. Bob, sitting on a different moving train, will describe the seats in his carriage as stationary but those in Alice's carriage as moving. Finally, Charlie, sitting comfortably on the station, will describe both sets of seats as moving.

But there's still a big question – is there anything that these three will agree about?

To measure velocity you need to choose a point of view

Velocity is a tricky thing to measure. One of the reasons it is tricky is that humans have no inbuilt velocity sensor. We do have inbuilt acceleration sensors, as a result of evolving and living on a mass that exerts gravitational forces on us. So one way to get a handle on velocity is to start with acceleration.

Acceleration is something we have a feel for: there are physical experiences that can be directly correlated with the physical quantity acceleration. (If you have any lingering doubts, try waggling your head, or jumping up and down, or go for a drive along a winding country lane.)

We've shown that acceleration and velocity are intimately connected. Acceleration tells velocity how to accumulate. An acceleration of 2 metre second^-2 simply means adding 2 metre second^-1 onto the existing velocity for each second of that acceleration. An acceleration of -2 metre second^-2 simply means subtracting 2 metre second^-1 from the existing velocity for each second of the acceleration. So one can see that a constant (and non-zero) acceleration will always be connected with a steady increase or decrease in velocity. If the acceleration is zero, then the velocity will neither increase or decrease: the velocity will be constant. These simple and necessary patterns allow you to make inferences in the reverse direction, from records of velocity to the acceleration connected with those velocities.

Graphical accumulations

One way of presenting a record of velocities is to use a scatter graph. In the case where the velocity is increasing (a positive acceleration) there will be one characteristic shape. The shape will be different if the acceleration is zero (the velocity is not changing), and different again if the velocity is decreasing (a negative acceleration).

The necessary connection between acceleration and velocity result in necessary connections between the shapes of the graphs representing a record of these values over time. So, you can move easily between the acceleration–time and velocity–time representations, so long as you remember that the connection between velocity and acceleration is that acceleration tells velocity how to change. The greater the acceleration, the greater the gradient of the velocity– time graph. If the acceleration is positive, then the gradient will also be positive (sloping upwards as time increases). If the acceleration is negative, then the gradient will also be negative (sloping downwards as time increases). If the acceleration is zero then the gradient will also be zero (no slope at all).

Velocity and changing position

Alice, Bob and Charlie are once again at the railway station (no rest for good behaviour). Charlie is on the platform and Alice is in the up-train whilst Bob is in the down-train.

The up-train moves past: Charlie notices that it gets 14 metre farther away from him each second. He might have written this as velocity of 14 metre inverse second. Next, he notices Bob on the down-train: Charlie notices that the separation between Bob and himself decreases by 8 metre inverse second. The direction is reversed, and to bring attention to this feature he records the velocity as -8 metre inverse second.

How will Alice and Bob record Charlie's velocity?

And what about Charlie_displaced, on another platform just along the way?

Surely he'll see the up-train gets 14 metre closer to him each second, so he'll record the velocity as -14 metre inverse second. But the down-train is also moving so that the separation between Bob and himself decreases by 8 metre inverse second. So he records the velocity as -8 metre inverse second.

Charlie and Charlie_displaced are consistent in taking velocity as change in positiontime, and yet they disagree about the velocities, even through they're recording values for the same trains, at the same time. There is something strange going on here, and we'd better pay attention to thinking about how we introduce the relationship between change in position, or displacement, and velocity.

Understanding velocity – it's essentially a vector

Treated as vectors, there's no problem.

The →v always determines the change in →s (the →d) in each second. There's an elegant simplicity here, if you use the most helpful formalism. (The difficulties in establishing consistency between Charlie and Charlie_displaced came with trying to simplify the vectors down to a single dimension and treat them as a signed quantity.)

displacement = velocity × duration (Or, in symbols, Δ d = v ×  Δ t).

Velocity has the same relationship to displacement (so to change in position) as acceleration does to velocity. The common pattern is one to keep in mind, as then everything you learn about the behaviour of one pair can easily be transferred to the second pair.

Where am I, and when am I there?

Acceleration, velocity and position have been introduced as descriptions of motion true at an instant. Questions such as When did it have that velocity? are best answered by giving a time of day – a clock reading. You're reporting the right-now status of the motion, not telling a story about how it has been over a period, or interval of time.

The tracks provided by a GPS unit provide a set of positions and clock times (at that instant – not before, and not after). They report where you were and when you were there (so an instantaneous record of position and clock reading). They locate you on a journey as a trail of events. Each event is a statement about where you were and when you were there. Such events are the basis of careful thinking about motion that is necessary to understand Einstein's theory of relativity.

It is possible that the trail records your position as unchanging for a number of different clock times: in that case we'd suggest that it's natural to say, for that duration, that you were stationary. The duration is then specified by the difference in clock times

(duration =  Δ clock time).

It's also possible that such a trail reports that your position changes over such a duration. We'd suggest that you call this difference in position displacement

(displacement =  Δ position)).

You are literally displaced, or repositioned. This change may or may not represent the actual journey you took – it all depends on how often the positions were recorded (so it may not be an accurate record of the distance you actually travelled). So it may not be a good idea to try and find the speed at which you were travelling from such a trail. You could find the velocity: →v = →d Δ t. But we suggest you leave this for post-16 study.

Suggestions for referring to quantities

Here are some suggestions as to how to refer to quantities, taking care to be consistent:

Teacher Tip: Clock readings (an instant) and position (instantaneous) together define an event – a location in space and time.

Teacher Tip: A series of (such) events – clock readings (an instant) and positions (instantaneous) – provides a trail, which can report a motion.

Teacher Tip: A period of time in such a motion is a duration, being the difference between two clock times.

Predictions – extrapolating to where you'll be, when you'll be there, and at what velocity you'll be travelling

Acceleration accumulates velocity, so that by finding out the acceleration and velocity now (at this instant, for this clock reading), you can gain some insight into the future – assuming some things don't change. This is the beginning of making predictive models, and is an essential part of physics. (The quality of the insight is often judged, to a significant degree, by the success it has in contributing to a predictive model.)

So here's a simple recipe, for predicting future velocities:

new value = old value + change

First calculate the change: Δ →v = →a ×  Δ t_{interval(A_B)}.

Then add the change to the old value: →v _final(AB) = →v _initial(AB) + →a × Δt .

You can carry out a similar process for displacement and velocity, so predicting changes in position. Remember that velocity accumulates displacement.

new value = old value + change

First calculate the change: Δ →d = →v ×  Δ t.

Then add the change to the old value: →d _final(AB) = →d _initial(AB) + →v × Δt .

When making predictions we'd suggest you use instantaneous values of acceleration, velocity, and position. Then run these forward from one instant (clock reading) to another instant (clock reading), so covering an interval, running from an initial instant to a final instant.

We'd suggest that you use →a, →v, →s or →d together with intervals of time ( Δ t) to project forward. Avoid averages.

Averages and reporting on journeys

When there's an actual track, such as might be recorded by a GPS unit recording a trail of events at a high frequency – that is, lots of (clock readings and position) records every second – then it's reasonable to suppose that it does map an actual journey, depicting rather accurately the distances covered.

In this case, avoid thinking with the vector terms (→a, →v, →s or even acceleration, velocity, position) and use the appropriate scalars: distance from journey beginning to journey end (from one instant (clock reading) to another instant (clock reading)) over the duration.

Then you can use speed = distanceduration, where speed is necessarily average speed.

You can make a connection from these averages to instantaneous speed (the magnitude of →v, written ||→v ||). You simply split the journey into smaller and smaller chunks.

Split the journey duration (from beginning to end) into many intervals (from initial to final). As the time intervals get shorter and shorter, so the speed you calculate will get closer and closer to ||→v ||.

So you can see how the instantaneous and average speeds are connected, and which is closest to the ||→v ||.

Distance and displacement

A similar thing happens to the values of distance and ||→d || as you chop the journey into smaller and smaller pieces. That should make sense if you remember the original caution about the distance not being, in general, equal to the ||→d ||.

(You can, in fact, see this averaging error in progress when you reduce the number of points in a GPS track. Any software that re-constitutes your track from the simplified trail does so by drawing straight lines between the (newly reduced) points. This straightens out the wiggles, so under-reporting the actual distance you covered.)

Where are you, and when are you there?

We've introduced the idea that looking out in space is looking back in time on several occasions throughout the SPT materials (for example, in the SPT: Motion topic, in the SPT: Earth in space topic and in the SPT: Radiations and radiating topic). This is dramatically obvious when the distances are huge and it takes the radiations an appreciable time to travel the huge distances (one light year is 9,460,800,000,000,000 metre).

Alice, based on Earth, doesn't see the Sun as it is now, but as it was 8 minutes ago – that's how long the information from the event happening there now takes to reach Alice. Reflected light from the Moon travels to the Earth in about 1 second. So, again, Alice only notices what's happening on the Moon one second after Bob (the astronaut on the moon) notices the event.

But we've also emphasised that the same (universal) connection between space and time applies locally: Closer to home light travels about 30 centimetre in one thousand millionth of a second – a foot per nanosecond – so even on the scale of a room, you only know about the past.

This has real consequences for Alice and Bob, who need to find ways to agree on the measurements for a particular process, even when they're not right next to it. Sitting them both beside the event (or even equal distances from it), and so having them co-moving with the event (it appears stationary to them), is one way of enforcing agreement. But if Alice and Bob are not travelling at the same velocity, they cannot both be at the start event of the process and at the end event of the process.

For many practical purposes this lack of being right there does not make much difference, as the time differences in the durations they record are tiny. But the greater the difference in the velocities of Bob and Alice, the greater the difference in displacements from the events, and so the greater the differences in the measurements. The distance from the event and the time at which you record it are necessarily linked by the universal speed (3 × 10⁸ metre inverse second).

The universal speed, which converts differences in distance to differences in time, at which all electromagnetic radiations (including light) travel, ensures that space and time will be entangled in this way. Measurements of space and time are not as distinct as we think. There is a measure that Bob and Alice will agree on, but it has spatial and temporal components.

It turns out that there is no universal clock, that all must agree on. Or, as Einstein reported his own thinking: It came to me that time was suspect.

So began an intellectual journey that ended with the special theory of relativity, all arising from some very careful thinking about velocity, and how people taking different points of view might record clock times and positions.

Accumulations set the connection between graphs

Physics is all about noticing, and then reusing, common patterns. You got a handle on velocity–time graphs by thinking about acceleration as the quantity which told velocity how to accumulate. By now, we hope you will not be surprised when we suggest getting a handle on displacement–time graphs by thinking about velocity as the quantity which tells displacement how to accumulate.

First consider discrete accumulations, to see the patterns in the two graphs.

Accumulating over smaller steps

One way of presenting a record of displacements is to use a scatter graph. You might consider several situations:

• The displacement is increasing (a positive velocity)  →  one characteristic shape.
• The displacement is not changing (the velocity is zero)  →  a different shape.
• The displacement is decreasing (a negative velocity)  →  yet another shape.

The necessary connection between velocity and displacement results in necessary connections between the shapes of the graphs representing a record of these values over time. So, you can move easily between the representations of velocity–time and displacement–time, so long as you remember that the connection between displacement and velocity is that velocity tells displacement how to change.

Here is a spelt-out summary of the connections:

• The greater the velocity, the greater the gradient of the displacement–time graph.
• If the velocity is positive, then the gradient will also be positive (sloping upwards as time increases).
• If the velocity is negative, then the gradient will also be negative (sloping downwards as time increases).
• If the velocity is zero then the gradient will also be zero (no slope at all).

Accumulating whilst the velocity changes

For a constant velocity the displacement increases by a fixed amount in each unit of time, and this varies as the value of the velocity changes. Therefore as the velocity varies, so the gradient of the displacement–time graph will vary. This gradient will be a curve. The curve can be concave or convex: it'll be set by the variations in the velocities. You might take a moment to figure out just why the curves have the shapes they do in the graphs on this page.

If the velocity is increasing then the increments will increase – a concave curve.

If the velocity is decreasing then the increments will get smaller – a convex curve.

Acceleration and velocity

A rather concise way of expressing the connection between velocity and accelerations is to say that the rate of change of velocity is the acceleration. (The rate of change of velocity is the increment or decrement of velocity in a unit of time.) This is exactly the converse of saying that acceleration tells velocity how to accumulate.

We think that talking about the accumulation – just adding on either positive or negative quantities – is much simpler than introducing the idea of rate of change. Rates of change are complex because of the multiple operations required to calculate them (subtraction and division) and hard because to be precise you need to move to the limit: the duration you divide by is a very short length of time (literally an infinitesimal duration). Adding small contributions (so accumulating over small durations) seems accessible, and is not a complex operation (we think addition is very simple).

The belief that accumulations is a better approach than rates underpins this approach to the study of motion.

More simply still, we're suggesting that addition is simpler than division: new value = old value − change.

Where the change is over some agreed duration, the more complicated route is: rate = value_new — value_oldtime interval.

And, of course, Δ t should be made very small, so that the duration is as close to zero as we can imagine.

Velocity and displacement

A similarly concise expression connects position and velocity: the rate of change of position is the velocity. (The rate of change of position is the increment or decrement of position in a unit of time.) Again, this is exactly the converse of saying that velocity tells position how to accumulate. We think that talking about the accumulation – just adding on either positive or negative quantities – is much simpler than introducing the idea of rate of change. This belief underpins the approach to the study of motion here developed.

Again, we're suggesting that addition is simpler than division.

Here is the accumulations version written out:

new displacement = old displacement + change due to velocity

Where the change due to the velocity is simply: change = velocity × chosen interval of time (or, in symbols s_new = v ×  Δ t).

The explicit recipe for accumulating displacement over the interval AB is therefore: s _final(AB) = s _initial(AB) + v × Δt .

Following the division route:

velocity = displacement_new — displacement_oldtime interval between displacements (or, in symbols v = s_new — s_old Δ t ).

Connecting acceleration, velocity, displacement

Here is a summary of both relationships, seen as accumulations:

• Acceleration determines change in velocity.
• Velocity determines change in position.

Here is a summary of both relationships, seen as rates:

• Acceleration is the rate of change of velocity.
• Velocity is the rate of change of position.

Going all the way

In this episode you've concentrated on the relationships between the measures of motion, exploring the connections between acceleration, velocity and displacement.

• Acceleration accumulates velocity.
• Velocity accumulates displacement.

In the first episode you concentrated on the causes of motion – that is, what got things going and otherwise changed their motion: A resultant force causes a change in motion. These eight words imply a lot of understanding: the whole of the SPT: Forces topic and the whole of the SPT: Motion topic are needed to really understand the phrase. Physics may be simple but it is quite subtle.

• Force causes acceleration.
• Mass impedes acceleration.

Now we simply put the three together to get a very powerful synthesis: a way of building predictive models of the world that are very reliable guides to action.

• forcemass sets acceleration
• Acceleration accumulates velocity.
• Velocity accumulates displacement.

Some quantities have direction: some do not

Some physical quantities are essentially directional. Force is one such. As soon as you describe a force, it's possible to ask about its direction. You can't answer that it has no particular direction, or any direction at all. These quantities are described using vectors – ordered sets of numbers, together with units.

Some physical quantities have no direction – that is, it makes no sense to attach a direction to the description. Energy is one such quantity. Such quantities are described with a single number, and usually a unit.

Descriptions of motion are best treated with vectors, and that is what we've suggested throughout; →a, →v, →s. These are all properties of an object (as seen from a chosen point of view), so it makes sense for the arrows to be linked to such an object, just as forces always act on an object.

Simplifications to a single dimension can lead to acceleration, velocity and position being treated as signed quantities. Such treatments can obscure the essential vector nature of the entities, and create difficulties (more on this in the Teaching and Learning Issues strand).

How many ordered numbers you need to define the quantity depends on the number of dimensions. In this world we normally use three dimensions.

→v = 3 metre / second2 metre / second −  1 metre / second

→s =  −  5 metre2 metre −  7 metre

Distance, position and displacement

Displacement is a vector quantity that refers to "how far out of place an object is"; it is the object's overall change in position.

That suggests that you ought to be rather careful to separate moving things from repositioning them, perhaps reserving moving for changes where you're interested in the journey, and not only in the end points.

If you are interested only in the end points then changing the position is equivalent to displacing the object.

Distance is a scalar quantity that refers to how much ground an object has covered during its motion.

If you undertake a trip or journey then you move – that is you traverse a distance. This may involve lots of wiggles, so the distance may not be equal to the magnitude of the displacement (||→s ||).

Acceleration, velocity, speed

Acceleration is a vector, so it needs an ordered set to describe it. One such set can include the magnitude of the acceleration. Other members of the set will then have to include the direction of the acceleration. As the world is three-dimensional, there will be three numbers, in this case the magnitude and two angles.

→a = magnitudeangleangle

You'd have to already have a pair of lines defined, from which to measure the angles.

You could use three other numbers. For example:

→a = acceleration in x directionacceleration in y directionacceleration in z direction

Again, here you need some agreed background to make this description intelligible – here the direction of the x, y and z axes.

In summary, →a is useful, and ||→a || can be useful, but there's no special name in general use.

Velocity is a vector, so it's best described by an ordered set of numbers and units. One such set can include the magnitude of the velocity. Other members of the set will then have to include the direction of the velocity. As the world is three-dimensional, there will be three numbers, in this case the magnitude and two angles.

→v = magnitudeangleangle

You'd have to already have a pair of lines defined, from which to measure the angles.

You could use three other numbers. For example:

→v = velocity in x directionvelocity in y directionvelocity in z direction

Again, here you need some agreed background to make this description intelligible – here the direction of the x, y and z axes.

In summary, →v is useful, and ||→v || can be useful, and it has an everyday name: speed.

The rate of separation sets the relative velocity

Reporting any velocities unambiguously depends on making explicit the point of view that you've chosen. So all velocities are relative velocities – that is they report rates of separation. Usually it's the separation between the observer (Alice, Bob or Charlie) and what they're reporting on.

However, sometimes you'll want to work out, from what Alice reports and what you record about Alice's velocity, the velocity of the object that Alice is reporting on. That's when you need to add together the two rates of separation – the two velocities (→v_Alice-Object and →v_Alice-You). Of course, you'll remember that these are vectors that you're adding.

The ability to switch points of view by finding relative velocities is a powerful way of thinking, and you'll use it again in episode 03 to link the conservation of momentum to Newton's first law.

Predicting circular and parabolic paths

Velocity vectors predict where something will end up after a short interval of time. Following a sequence of such predictions maps out a route that you'd expect the object to follow.

Two interesting patterns that arise are where the vectors change in a systematic way with time. Changes in the velocity vector over time are, of course, the results of an acceleration.

There could be any number of patterns of change in the velocity vectors.

These two are of particular interest because of the resulting patterns of motion.

• The change in the velocity is at right angles to the existing vector.
• The change in velocity vector is always in one direction, and is a result of uniform acceleration.

These produce circles and parabolas, both of which are rather common motions.

A repeated pattern

A central idea here, repeated often, is that accumulations over time lead to changes. To understand the structure of these changes clearly we need to be precise, and to use vectors.

The three central ideas in this episode are acceleration (→a), velocity (→v), position (→s).

Time is not used; instead we suggest clock time, duration and interval.

As you shift from vector to scalar descriptions, so you'll need to be careful that you don't drop or obscure essential elements. So we've suggested a restricted role for (average) speed, distance and duration.

Similarly, we've suggested that a recognition that all movement is relative movement is the key to avoiding ambiguity.

However, some things (for example, mass, force and acceleration) don't change as you switch point of view by adopting a new velocity – they're invariant – which provides a sound lead in to relativistic thinking, from Galileo to Einstein. Charlie, Alice and Bob have always agreed about force, acceleration and mass. That's because they've been helping you explore a Newtonian world. (In an Einsteinian world, they'd not even agree about durations and displacements, but that's for later study.)