PHYSICAL EXAMPLES OF VECTORS

1. Introduction

The properties of vectors which have been discussed in the previous sections can all be applied to physical quantities which are vectors. In general, all physical quantities which require a magnitude and a direction in order to be fully described are vectors. Such physical quantities are distinguished from scalar quantities which have magnitude only.

2. Distance and displacement

DISPLACEMENT is a vector quantity which represents the difference in the position of two points. It is given the symbol s and has units of metres (m) in a specified direction.

Displacement must be distinguished from the scalar quantity DISTANCE, which has the units of metres. The difference between distance and displacement is illustrated by the following problem:

A man wishes to empty a cylindrical water tank with a diameter of 10 m. The outlet valve is on the opposite side of the tank, due north of him. We would like to know (i) the displacement of the valve from his present location, and (ii), the minimum distance he must walk in order to open the valve.

Answer: The displacement of the valve from the man is 10 m North, which is twice the radius of the circular tank).The minimum walking distance is halfway around the tank, which is π x radius = 3.14 x 5 (m) = 15.7 m.

3. Speed and velocity

SPEED is a scalar quantity defined as the RATE OF CHANGE OF DISTANCE. The units of speed are metres per second (m·s-1).

The AVERAGE SPEED, v, may be calculated by dividing the distance travelled, Δx, by the time taken, Δt, to cover the distance, for example,

Since the speed may vary during the journey, it is useful to consider the speed over a very short time interval, Δt (pronounced "delta tea"), during the journey i.e., v = Δx/Δt, where Δx is the distance travelled in the very short time Δt. (Strictly speaking, one should be concerned with instantaneous speed).

VELOCITY is a vector quantity defined as the RATE OF CHANGE OF DISPLACEMENT. It is given the symbol v and has units of metres per second (m·s-1) in a specified direction. AVERAGE VELOCITY may be defined as the total displacement divided by the time taken to make that displacement, that is, v = Δx/Δt.

Instantaneous velocity is defined in a similar way to that of instantaneous speed).

4. Acceleration

ACCELERATION is a vector quantity defined as the RATE OF CHANGE OF VELOCITY. It is given the symbol a and it has units of meters per second squared (m·s-2).

AVERAGE OF ACCELERATION, a is calculated by dividing the change in velocity Δv that occured during a time interval, Δt, by that time interval Δt. For example, if the velocity of an object changed from 3 m·s-1 to 5 m·s-1 in 5 seconds, then the average acceleration can be calculated:

(See also instantaneous acceleration)

Objects in motion undergo acceleration if either the speed or the direction of motion changes. Motion in which the direction does NOT change is called RECTILINEAR MOTION (i.e., the object moves in a straight line - see here what is meant by non-rectilinear motion). If the instantaneous acceleration is the same throughout a given time interval, then the object is said to be UNIFORMLY ACCELERATED. In this case, the velocity changes by equal amounts in equal intervals of time.

If a body which has an initial velocity, vi, undergoes an acceleration a for a time Δt, we can see from the definition of acceleration that the final velocity vf will be given by vf = vi + aΔt

If the acceleration is UNIFORM, then the average velocity, v, over the time interval is:

which is the mean of the initial and final velocities. This is only true if the acceleration over the interval, t, is uniform, in which case is also the instantaneous velocity at the midpoint of the interval, that is, Δt/2.

5. Experimental determination of s, v and a

In school experiments the change in displacement, velocity, and acceleration of bodies in rectilinear motion can be measured as a function of time by attaching the bodies to a paper tape and having the tape passing through a ticker timer:

The ticker timer produces a series of dots on the paper tape at equal, known time intervals, T. These time intervals are related to the frequency, f, of the device, by:


The change in displacement of the object as a function of time can be measured directly from the tape. (The experimenter must choose the origin appropriately).

The average velocity of the object in a time interval can be calculated by measuring the change in displacement during the interval and dividing it by the time interval:

If the object is undergoing uniform acceleration, then the spacing between the dots will increase:

The average acceleration may be calculated by determining the average velocity at two different times and then applying the formula a = Δv/Δt.


6. Force

If a coiled spring is stretched or compressed, it is said that a force is acting on the spring.

Spring under compression and stretching

This definition of force is used in the spring balance which is commonly used to measure forces.

Spring balance

Alternatively, if a force is applied to an object, the object will accelerate in proportion to the magnitude of the force and in the direction of the applied force:

Since force has a magnitude and a direction, it is a vector. The unit of force is the NEWTON (N) in a specified direction. A force of 1 N on a mass of 1 kg will cause that mass to accelerate at 1 m·s-2:

If several forces are applied to an object and the object remains stationary, or if the object continues to move with uniform velocity, the forces are said to be in EQUILIBRIUM. Forces in equilibrium add to produce a resultant of zero.

If several forces, not in equilibrium, act on a body, the force which is required to produce equilibrium is called the EQUILIBRANT:





7. Additional questions


Instantaneous speed

INSTANTANEOUS SPEED may be rigorously defined as

which is the value of Δx/Δt as Δt becomes infinitesimally small, that is, Δt tends to a value of zero, but does not attain it. Δx here describes the change in the scalar quantity, distance.

This is the value of v which is measured, for example, with a speedometer.


Instantaneous velocity

INSTANTANEOUS VELOCITY may be rigorously defined as

which is the value of Δx/Δt as Δt becomes infinitesimally small, that is, Δt tends to a value of zero, but does not attain it. Δx here describes the change in the vector quantity, displacement.

This is the value of v which is measured, for example, with a speedometer.


Instantaneous acceleration

INSTANTANEOUS ACCELERATION may be rigorously defined as

which is the value of Δv/Δt as Δt becomes infinitesimally small, that is, Δt tends to a value of zero, but does not attain it.


Non-rectilinear motion

The velocity changes when the direction of motion changes, even if the speed remains constant. For example, if a car is travelling around a circle at a constant speed, its direction and hence its velocity is changing continuously.

For example, a car starts at a point A, and takes 30 s to complete a circuit of radius 20 m, and then stops at point A. (Click here to see an animation.)