Vectors
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Vectors and scalars
Scalars
A scalar quantity can be described fully by stating its magnitude (size). Examples of scalar quantities are:
The quantities we write as ,
or
are all scalars.
Vectors
A vector is a quantity that is not fully described by stating its magnitude.
Forces are often thought of as a push or a pull.
Force is a vector quantity. Vectors possess a magnitude and a direction – both properties are required to describe the vector.
There are several vector quantities including:
- displacement (the distance and direction from where you started to where you finished)
- velocity (like speed, but in a certain direction. Velocity = Displacement ÷ Time),
- acceleration (the change in velocity per second, in a certain direction)
- force (to move an object or slow it down, a force must be applied to an object in a certain direction)
Displacement is a distance and a direction, eg south.
Velocity is a speed and a direction, eg on bearing 055 (55º East of North).
The term acceleration can refer to a scalar acceleration or an acceleration vector. So far, we have only met scalar acceleration, eg . Vector acceleration is a scalar acceleration and a direction, eg
to the right. Treat acceleration as a vector when there is another vector quantity, such as velocity or force, involved in a question.
Forces need a size and direction, eg to the left.
The relationship between distance, speed, and acceleration can be applied to displacement, velocity, and acceleration.
For scalars::
For vectors::
We also know that for scalars:
and for vectors:
Acceleration can be described as a vector or scalar depending upon how we determine it. In strict physics terms acceleration is a vector quantity.
Adding vectors
When adding vector quantities remember that the directions have to be taken into account.
The result of adding vectors together is called the resultant.
In problems, vectors may be added together by scale diagram or mathematically.
You should notice two things:
- Average speed and average velocity can have different magnitudes.
- The displacement and average velocity vectors point in the same direction.
Any vector quantities can be added together like this. It doesn’t matter if it is force or velocity, the vectors can still be added together the same way.
Useful terms for describing motion
The following terms are useful for describing and calculating different types of motion:
- average speed
- instantaneous speed (normally referred to as ‘speed’)
- acceleration
Average speed
Average speed is defined by the mathematical relationship shown below.
Average speed is measured in metres per second ().
Calculating the speed of a car
Instantaneous speed
Instantaneous speed is the speed of an object at a particular moment in time. It is measured in metres per second (), ie not over a long distance or long time period. Usually in the lab, instantaneous speeds are measured over a distance of a few centimetres in time of tenths of a second.
To do this in the classroom, a short piece of card is usually attached to a vehicle (distance = width of card) and the time is measured using a computer and light gate (time = time taken for card to pass through the light gate).