Ideas about speed

Ideas about speed are part of the language and experience of most pupils. When asked about the speed limit on the motorway pupils will readily use the language of 70 miles per hour. Speeding motorists are recorded on police cameras and reported in the news. Speed records for the fastest bird or animal, car or plane hold pupils' attention.

But there are subtleties. What I (Charlie) record as passing at 13 metre / second, you (Alice) may record as stationary (so a speed of 0). The reason may be simple: Alice is driving along in her car, and Charlie is standing on an overhead footbridge. The recording is a record of the speed of Alice's handbag, on the front seat of the car.

Alice notices no movement from her point of view: the handbag is at all times the same number of metres away from her. That's why she records the speed as 0. If Alice says that's at rest, she means nothing more or less than it's just moving with her.

Charlie notices the handbag getting 13 metre closer to the shadow of the bridge on the road every couple of seconds. That's why he records the speed as 13 metre / second.

Now consider Bob, cycling along at half of the car's speed, and overtaken by Alice. What will he record for the speed of that handbag?

Bob won't record what Alice does, nor what Charlie does. Each individual with a unique point of view will record a different speed. These will only agree with each other if they are moving together. Duncan, cycling along with Bob, agrees with Bob, for example. Elizabeth, sitting in the rear seat of the car, agrees with Alice. Fayed, standing with Charlie, agrees with him.

Suggestion: to be clear, make sure your point of view is explicit. This gets more important as the situations get more involved, and as you expect more careful thinking. Thinking about Einstein's theory of relativity starts here.

So far as we can tell, there is no absolute point of view. No absolute at rest. At rest means staying the same distance from me.

What exactly do we mean by speed?

Speed is about movement, about travelling. But speed is not just about a distance travelled or a time taken. It is a way of reporting a rate of progress. It specifies how much distance is covered during a particular time interval. That is the increasing or decreasing separation between Alice and the object that Alice is reporting on.

Perhaps the simplest definition is:

Alice: Speed is the distance moved per second.

From this definition we can see that metres per second (metre / second) is a useful unit for speed. However, Alice might use any unit of distance and any unit of time. Here are a few examples:

• A caterpillar might have a speed of 4 millimetre / second.
• Your fingernails might grow at a speed of 2 millimetre each week.
• A jet fighter might travel at a speed of 1000 kilometre in each hour.

What is common to these examples is the idea of a rate of progress – two points move so that their separation changes with time. One or more points is re-positioned. The distance between them alters as time ticks by.

You will find in physics textbooks formal definitions such as:

Speed is the rate of change of distance with time

Teacher Tip: We'd suggest that you stick with standard units of metres and seconds in the early stages, avoiding quirky favourites such as furlongs per fortnight.

Converting from one set of units to another often involves arithmetic that may serve to obscure rather than to reveal.

Finding out speed from a distance and a time

A car speedometer tells us the speed of the car at a particular instant, giving a prediction for the number of metres closer to Charlie – standing ahead on the roadside – it'll be getting in each second. If we continue to convert time to distance at this rate, then we'll know how many metres it is closer to Charlie. A constant speed increments the distance at a fixed rate – the instruction is to add so many metres in each second. Run that accumulation for the number of seconds and predict where you will be.

Back to the car seat: Just look down and you know straight away how fast you are going, at least over the road. Such a speed is called an instantaneous speed. It is the speed at that particular instant. Even this is not strictly instant – but it's quite close as the change in distance is measured over a very short duration. Shorten the duration and you get closer and closer to instantaneous.

Just like speedometers, genuine instantaneous speed meters are rare (more in the SPT: Radiations and radiating topic). For many experimenters, speed is a quantity which needs to be calculated using the relationship:

speed = distance travelledtime taken The units of speed, for example metre/second (m/s) or kilometre/hour (km/hr), reflect the idea that speed is expressed as a distance divided by a time. When introducing pupils to measuring times and distances to find out speeds it is a good idea to select examples where the speed is constant. A body travelling at constant speed covers equal distances in equal times. Walking down an empty corridor is an example of something moving at a constant speed.

Calculating average speed

A car speedometer reports the speed of the car at (near enough) a particular instant. Just look down and you know straight away how fast you are going. Such a speed is called an instantaneous speed. It is the speed at that particular instant. Speedometers rarely make an appearance in school science. For classroom experimenters, speed is a quantity which needs to be calculated using the relationship:

speed = distance travelledtime taken

It is important to recognise that the calculated average speed does not represent the speed of a moving object at some particular instant.

Mary cycles to school. She travels 900 metre from her home to the school bike shed. She locks her bike and then walks the last 120 metre to the school building. The cycle ride takes two minutes and the walk takes one minute.

When cycling her (constant) speed was 7.5 metre / second. When walking her (constant) speed was 2 metre / second.

Let's calculate Mary's average speed.

average speed = distance travelledtime taken

Which, in her case is 1020 metre180 second, or 5.7 metre1 second.

So Mary's average speed is 5.7 metre / second.

Calculations based on different measurements

As the interval over which an average speed is measured becomes shorter and shorter, so this speed becomes closer and closer to the instantaneous speed.

What you see on the speedometer of a car is the speed at that instant or moment – the instantaneous speed. It's the speed right now. One way to find this instantaneous speed is to measure the rate of rotation of the wheels. Modern electronic devices allow accurate measurement of short time intervals and a sensor can measure the angle during these very short time intervals, effectively giving an instantaneous speed.

For the most part, however, we measure longer journeys: the distance travelled over longer time intervals from several seconds to minutes or even hours. The resulting calculation gives an average speed. We do not assume that the car maintained a constant speed during the time.

Average speed calculated at the end of a journey can tell you that:

• The athlete who completed the 800 metre race in 160 second had an average speed of 5 metre / second.
• The car that completed a 20 mile journey in 30 minutes had an average speed of 40 miles per hour.

It is common to use the term speed rather than average speed. Many teachers drop the term average thinking it may add an additional level of difficulty. We would nevertheless recommend that you encourage your pupils to make the distinction between instantaneous speeds and average speeds wherever possible. One way to do this is to restrict the use of average speed to refer to a whole journey. (Later you might relax this to apply to longish legs within a single journey, so treating them as a series of mini-journeys). Instantaneous speeds can, by contrast, then be right now measurements, during the journey. As the journeys get broken into shorter and shorter legs, so average speeds become more and more indistinguishable from instantaneous speeds. Nevertheless, it's a good way to keep the functions of the two measurements separate, until a more sophisticated understanding is worth developing (more in the SPT: Force and motion topic).

Distinguishing between speed and velocity

In the world of the sciences a clear distinction is made between the terms speed and velocity. In the 11–14 curriculum this distinction is not needed and our advice would be to stick with the more familiar term speed. Should you want to find out more, read on.

Speed and velocity are often used as if they were the same quantity and indeed they are measured in the same units (metre/ second or kilometre / hour). The key difference is that the speed of an object tells us about how quickly it is moving, while the velocity specifies how fast it is moving in a certain direction.

This being the case:

• Speeds are scalar quantities which have magnitude only (specified with a single number),

while

• Velocities are vector quantities for which both magnitude and direction must be given (specified with two or more numbers).

The upshot is that scalars can be described by a single quantity (number plus unit), whereas vectors need more than one such quantity, and these form an ordered set (so the position of the numbers is important – don't mix up the direction with the magnitude). These kinds of ordered sets of quantities are very useful in the sciences, which is why many quantities are best described using vectors. But these really begin to be useful when children study physics beyond the current level – starting with that normally studied at 14–16 years old.

For example, when talking about a car's velocity, we might say that it is travelling at 30 mph due north.

A trolley may be described as having a velocity of 10 metre / second to the right. This may seem to be a trivial difference, but it becomes important in situations where direction matters.

A pupil walking backwards and forwards across a room may have a constant speed of 2 metre / second. However, speed gives no indication of the direction in which the pupil is walking. Using velocity enables us to state that the pupil has a velocity of +2 metre / second when walking to the right and a velocity of -2 metre / second when walking to the left. This gives a more precise account of the motion of the pupil.

The designation of one direction as positive and the other direction as negative is arbitrary. Often up is chosen as positive (down is negative) and right is chosen as positive (left is negative). It's a good idea to be very clear about what you've chosen, as assuming that the direction in which something is initially moving is positive, without noticing your assumption, can lead to difficulties later.

Differences recorded as a physical quantity

Relative movement, happening over duration, can be recorded as speed. Speed determines the rate of change position, so how quickly the distance changes with time.

Speed is a physical quantity so you must always quote a number and a unit, never just a number.