1) Distance and Displacement

Suppose a dog walks round a rectangle from A to B. The total distance travelled by the dog is 17 m, however the displacement of the dog is the direct distance from where the dog started to where he ended. Thus the displacement of the dog is 13m (23⁰) (ie 13m relative to where the dog started). Also, the moon travels a great distance round the Earth each month but its displacement is zero. Note: We speak of distance travelled by a body and the displacement of a body. This is the difference between distance and displacement.

2) Speed and Velocity

The speed of a body is simply the distance the body covers in unit time (second, minute, hour). However, the average velocity of a body indicates the distance covered in a unit time in a given direction.



To help distinguish speed and velocity further consider the following. If a car had an average speed of 20 ms^-1 for a one hour journey, it is unlikely the car travelled at a constant speed of 20ms^-1 for the full hour, indeed it may have travelled at that speed for only brief instances. A similar argument holds with the direction part of an average velocity. Although our plane had travelled in a Northerly direction and its final velocity is given as North, at only one point (extreme West) was it travelling due North. In fact over the 200 s we say the plane covered 0.1 km every second in a Northerly direction. (even if it only headed due North only once and never at 0.1 kms^-1. You should be able to see how these concepts are useful for navigators in ships and aeroplanes. The velocity and speed of a body will have the same magnitude only if the motion is in one dimension and the body does not reverse its direction. Even though the Earth travels round the Sun at a great speed, its average velocity is zero.

You are invited to calculate the speed and velocity of the plane after 300 s, if it continues in the same circular route.

3) Scalars and Vectors

Scalars are physical quantities that have magnitude (size) only and thus can be defined by one number (with unit). For example the speed of the aeroplane was 0.16 m^-1 and thus this single value completely defined the physical quantity we call speed. Speed is a Scalar. Vectors are physical quantities that have magnitude and direction and thus require two values to completely describe them. For example the velocity of our plane was 0.1 ms^-1, North. and thus two values are required to define the physical quantity we call velocity. The angle provides information about direction. Velocity is a vector. The table below lists some of the scalars and vectors we work with in this course.

Scalar	Vector
Distance	Displacement
Speed	Velocity
Mass	Weight (force)
Time	Acceleration
Temperature	Impulse

4) Scale diagrams and Resultants

Vectors are represented by arrows. The magnitude of the arrow is its magnitude and the arrow indicates its direction The resultant vector is the sum of two or more vectors. If the vectors span two (or more) dimensions then direct addition is not possible and an accurate scale drawn must be drawn. The following example illustrates this. Example: It is a calm day at sea and a ship is sailing through the water. The ship travels with a bearing 060^o for 10km, the ship changes direction and travels with a bearing 110^o for 5 km, the ship again changes direction and travels a further 8km on a bearing of 300^o. Find the the resultant displacement vector, ie locate the position of the ship relative to where it started.

5) Resultant Force

The resultant force vector is calculated in the same way as the resultant displacement vector, but they are thought of differently. Where the resultant displacement vector locates the position of a body relative to where it started, the resultant force vector is the single force that is equivalent to the two or more force vectors that made it up. The following example illustrates this.
Ex: Two tug boats are pulling an oil platform out to sea. The boats each exert a force of 1.0x10⁵ N at an angle of 20^o as shown. Find the resultant force vector, ie find the single force vector that is equivalent to the two vectors that made it up.

6) Rectangular Components of a Vector

The resultant vector is generally the sum of many vectors. When presented with a vector it is often useful to calculate its rectangular components. These rectangular components can be represented as vectors and their magnitude can be measured directly or calculated using simple trigometry. To find the direction of these vectors we use the head to toe method of vector addition The vector below represents a velocity vector and its rectangular components are called i) Horizontal component of velocity (ie in our example this is; 43.3 ms^-1) ii) Vertical component of velocity (ie in our example this is; 25.0 ms^-1)


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Higher Physics .. Unit 1 Mechanics and Properties of Matter .. 1.1 Vectors .. BR (2011)