By: Sunil Singh
Online: < http://cnx.org/content/col10348/1.29>
This selection and arrangement of content as a collection is copyrighted by Sunil Singh.
It is licensed under the Creative Commons Attribution License: http://creativecommons.org/licenses/by/2.0/
Collection structure revised: 2008/09/28
For copyright and attribution information for the modules contained in this collection, see the " Attributions" section at the end of the collection.
Table of Contents
Chapter 1. Motion
Motion is a state, which indicates change of position. Surprisingly, everything in this world is
constantly moving and nothing is stationary. The apparent state of rest, as we shall learn, is a
notional experience confined to a particular system of reference. A building, for example, is at
rest in Earth’s reference, but it is a moving body for other moving systems like train, motor,
airplane, moon, sun etc.
Figure 1.1. Motion of an airplane
The position of plane with respect to the earth keeps changing with time.
Motion of a body refers to the change in its position with respect to time in a given frame of
A frame of reference is a mechanism to describe space from the perspective of an observer. In
other words, it is a system of measurement for locating positions of the bodies in space with
respect to an observer (reference). Since, frame of reference is a system of measurement of
positions in space as measured by the observer, frame of reference is said to be attached to the
observer. For this reason, terms “frame of reference” and “observer” are interchangeably used to
In our daily life, we recognize motion of an object with respect to ourselves and other stationary
objects. If the object maintains its position with respect to the stationary objects, we say that the
object is at rest; else the object is moving with respect to the stationary objects. Here, we conceive
all objects moving with earth without changing their positions on earth surface as stationary
objects in the earth’s frame of reference. Evidently, all bodies not changing position with respect
to a specific observer is stationary in the frame of reference attached with the observer.
We require an observer to identify motion
Motion has no meaning without a reference system.
An object or a body under motion, as a matter of fact, is incapable of identifying its own motion.
It would be surprising for some to know that we live on this earth in a so called stationary state
without ever being aware that we are moving around sun at a very high speed - at a speed faster
than the fastest airplane that the man kind has developed. The earth is moving around sun at a
speed of about 30 km/s (≈ 30000 m/s ≈ 100000 km/hr) – a speed about 1000 times greater than
the motoring speed and 100 times greater than the aircraft’s speed.
Likewise, when we travel on aircraft, we are hardly aware of the speed of the aircraft. The state of
fellow passengers and parts of the aircraft are all moving at the same speed, giving the impression
that passengers are simply sitting in a stationary cabin. The turbulence that the passengers
experience occasionally is a consequence of external force and is not indicative of the motion of
It is the external objects and entities which indicate that aircraft is actually moving. It is the
passing clouds and changing landscape below, which make us think that aircraft is actually
moving. The very fact that we land at geographically distant location at the end of travel in a short
time, confirms that aircraft was actually cruising at a very high speed.
The requirement of an observer in both identifying and quantifying motion brings about new
dimensions to the understanding of motion. Notably, the motion of a body and its measurement is
found to be influenced by the state of motion of the observer itself and hence by the state of
motion of the attached frame of reference. As such, a given motion is evaluated differently by
different observers (system of references).
Two observers in the same state of motion, such as two persons standing on the platform, perceive
the motion of a passing train in exactly same manner. On the other hand, the passenger in a
speeding train finds that the other train crossing it on the parallel track in opposite direction has
the combined speed of the two trains
. The observer on the ground, however, find them
running at their individual speeds v 1 and v 2.
From the discussion above, it is clear that motion of an object is an attribute, which can not be
stated in absolute term; but it is a kind of attribute that results from the interaction of the motions
of the both object and observer (frame of reference).
Frame of reference and observer
Frame of reference is a mathematical construct to specify position or location of a point object in
space. Basically, frame of reference is a coordinate system. There are plenty of coordinate
systems in use, but the Cartesian coordinate system, comprising of three mutually perpendicular
axes, is most common. A point in three dimensional space is defined by three values of
coordinates i.e. x,y and z in the Cartesian system as shown in the figure below. We shall learn
about few more useful coordinate systems in next module titled " Coordinate systems in physics
Figure 1.2. Frame of reference
A point in three dimensional space is defined by three values of coordinates
We need to be specific in our understanding of the role of the observer and its relation with frame
of reference. Observation of motion is considered an human endeavor. But motion of an object is
described in reference of both human and non-human bodies like clouds, rivers, poles, moon etc.
From the point of view of the study of motion, we treat reference bodies capable to make
observations, which is essentially a human like function. As such, it is helpful to imagine
ourselves attached to the reference system, making observations. It is essentially a notional
endeavor to consider that the measurements are what an observer in that frame of reference would
make, had the observer with the capability to measure was actually present there.
Earth is our natural abode and we identify all non-moving ground observers equivalent and at rest
with the earth. For other moving systems, we need to specify position and determine motion by
virtually (in imagination) transposing ourselves to the frame of reference we are considering.
Figure 1.3. Measurement from a moving reference
Take the case of observations about the motion of an aircraft made by two observers one at a
ground and another attached to the cloud moving at certain speed. For the observer on the ground,
the aircraft is moving at a speed of ,say, 1000 km/hr.
Further, let the reference system attached to the cloud itself is moving, say, at the speed of 50
km/hr, in a direction opposite to that of the aircraft as seen by the person on the ground. Now,
locating ourselves in the frame of reference of the cloud, we can visualize that the aircraft is
changing its position more rapidly than as observed by the observer on the ground i.e. at the
combined speed and would be seen flying by the observer on the cloud at the speed of 1000 + 50
= 1050 km/hr.
We need to change our mind set
The scientific measurement requires that we change our mindset about perceiving motion and its
scientific meaning. To our trained mind, it is difficult to accept that a stationary building standing
at a place for the last 20 years is actually moving for an observer, who is moving towards it. Going
by the definition of motion, the position of the building in the coordinate system of an
approaching observer is changing with time. Actually, the building is moving for all moving
bodies. What it means that the study of motion requires a new scientific approach about
perceiving motion. It also means that the scientific meaning of motion is not limited to its
interpretation from the perspective of earth or an observer attached to it.
Figure 1.4. Motion of a tree
Motion of the tree as seen by the person driving the truck
Consider the motion of a tree as seen from a person driving a truck (See Figure above) . The tree is undeniably stationary for a person standing on the ground. The coordinates of the tree in the
frame of reference attached to the truck, however, is changing with time. As the truck moves
ahead, the coordinates of the tree is increasing in the opposite direction to that of the truck. The
tree, thus, is moving backwards for the truck driver – though we may find it hard to believe as the
tree has not changed its position on the ground and is stationary. This deep rooted perception
negating scientific hard fact is the outcome of our conventional mindset based on our life long
perception of the bodies grounded to the earth.
Is there an absolute frame of reference?
Let us consider following :
In nature, we find that smaller entities are contained within bigger entities, which themselves are
moving. For example, a passenger is part of a train, which in turn is part of the earth, which in turn
is part of the solar system and so on. This aspect of containership of an object in another moving
object is chained from smaller to bigger bodies. We simply do not know which one of these is the
ultimate container and the one, which is not moving.
These aspects of motion as described in the above paragraph leads to the following conclusions
about frame of reference :
"There is no such thing like a “mother of all frames of reference” or the ultimate container, which
can be considered at rest. As such, no measurement of motion can be considered absolute. All
measurements of motion are, therefore, relative."
Nature displays motions of many types. Bodies move in a truly complex manner. Real time
motion is mostly complex as bodies are subjected to various forces. These motions are not simple
straight line motions. Consider a bird’s flight for example. Its motion is neither in the straight line
nor in a plane. The bird flies in a three dimensional space with all sorts of variations involving
direction and speed. A boat crossing a river, on the other hand, roughly moves in the plane of
Motion in one dimension is rare. This is surprising, because the natural tendency of all bodies is to
maintain its motion in both magnitude and direction. This is what Newton’s first law of motion
tells us. Logically all bodies should move along a straight line at a constant speed unless it is
acted upon by an external force. The fact of life is that objects are subjected to verities of forces
during their motion and hence either they deviate from straight line motion or change speed.
Since, real time bodies are mostly non-linear or varying in speed or varying in both speed and
direction, we may conclude that bodies are always acted upon by some force. The most common
and omnipresent forces in our daily life are the gravitation and friction (electrical) forces. Since
force is not the subject of discussion here, we shall skip any further elaboration on the role of
force. But, the point is made : bodies generally move in complex manner as they are subjected to
Figure 1.5. Real time motion
A gas molecule in a container moves randomly under electrostatic interaction with other molecules
Nevertheless, study of motion in one dimension is basic to the understanding of more complex
scenarios of motion. The very nature of physical laws relating to motion allows us to study motion
by treating motions in different directions separately and then combining the motions in
accordance with vector rules to get the overall picture.
A general classification of motion is done in the context of the dimensions of the motion. A
motion in space, comprising of three dimensions, is called three dimensional motion. In this case,
all three coordinates are changing as the time passes by. While, in two dimensional motion, any
two of the three dimensions of the position are changing with time. The parabolic path described
Figure 1.6. Two dimensional motion
A ball thrown at an angle with horizon is described in terms of two coordinates x and y
One dimensional motion, on the other hand, is described using any one of the three coordinates;
remaining two coordinates remain constant through out the motion. Generally, we believe that one
dimensional motion is equivalent to linear motion. This is not further from truth either. A linear
motion in a given frame of reference, however, need not always be one dimensional. Consider the
motion of a person swimming along a straight line on a calm water surface. Note here that
position of the person at any given instant in the coordinate system is actually given by a pair of
coordinate (x,y) values (See Figure below).
Figure 1.7. Linear motion
Description of motion requires two coordinate values
There is a caveat though. We can always rotate the pair of axes such that one of it lies parallel to
the path of motion as shown in the figure. One of the coordinates, y y1 is constant through out the
motion. Only the x-coordinate is changing and as such motion can be described in terms of x-
coordinate alone. It follows then that all linear motion can essentially be treated as one
Figure 1.8. Linear and one dimensional motion
Choice of appropriate coordinate system renders linear motion as one dimensional motion.
Kinematics refers to the study of motion of natural bodies. The bodies that we see and deal with in
real life are three dimensional objects and essentially not a point object.
A point object would occupy a point (without any dimension) in space. The real bodies, on the
other hand, are entities with dimensions, having length, breadth and height. This introduces
certain amount of complexity in so far as describing motion. First of all, a real body can not be
specified by a single set of coordinates. This is one aspect of the problem. The second equally
important aspect is that different parts of the bodies may have path trajectories different to each
When a body moves with rotation (rolling while moving), the path trajectories of different parts
of the bodies are different; on the other hand, when the body moves without rotation (slipping/
sliding), the path trajectories of the different parts of the bodies are parallel to each other.
In the second case, the motion of all points within the body is equivalent as far as translational
motion of the body is concerned and hence, such bodies may be said to move like a point object. It
is, therefore, possible to treat the body under consideration to be equivalent to a point so long
rotation is not involved.
For this reason, study of kinematics consists of studies of :
1. Translational kinematics
2. Rotational kinematics
A motion can be pure translational or pure rotational or a combination of the two types of motion.
The translational motion allows us to treat a real time body as a point object. Hence, we freely
refer to bodies, objects and particles in one and the same sense that all of them are point entities,
whose position can be represented by a single set of coordinates. We should keep this in mind
while studying translational motion of a body and treating the same as point.
1.2. Coordinate systems in physics*
Coordinate system is a system of measurement of distance and direction with respect to rigid
bodies. Structurally, it comprises of coordinates and a reference point, usually the origin of the
coordinate system. The coordinates primarily serve the purpose of reference for the direction of
motion, while origin serves the purpose of reference for the magnitude of motion.
Measurements of magnitude and direction allow us to locate a position of a point in terms of
measurable quantities like linear distances or angles or their combinations. With these
measurements, it is possible to locate a point in the spatial extent of the coordinate system. The
point may lie anywhere in the spatial (volumetric) extent defined by the rectangular axes as shown
in the figure. (Note : The point, in the figure, is shown as small sphere for visual emphasis only)
Figure 1.9. A point in the coordinate system
A distance in the coordinate system is measured with a standard rigid linear length like that of a
“meter” or a “foot”. A distance of 5 meters, for example, is 5 times the length of the standard
length of a meter. On the other hand, an angle is defined as a ratio of lengths and is dimensional-
less. Hence, measurement of direction is indirectly equivalent to the measurement of distances
The coordinate system represents the system of rigid body like earth, which is embodied by an
observer, making measurements. Since measurements are implemented by the observer, they (the
measurements in the coordinate system) represent distance and direction as seen by the observer.
It is, therefore, clearly implied that measurements in the coordinates system are specific to the
state of motion of the coordinate system.
In a plane language, we can say that the description of motion is specific to a system of rigid
bodies, which involves measurement of distance and direction. The measurements are done, using
standards of length, by an observer, who is at rest with the system of rigid bodies. The observer
makes use of a coordinate system attached to the system of rigid bodies and uses the same as
reference to make measurements.
It is apparent that the terms “system of rigid bodies”, “observer” and “coordinate system” etc. are
similar in meaning; all of which conveys a system of reference for carrying out measurements to
describe motion. We sum up the discussion thus far as :
1. Measurements of distance, direction and location in a coordinate system are specific to the
system of rigid bodies, which serve as reference for both magnitude and direction.
2. Like point, distance and other aspects of motion, the concept of space is specific to the
reference represented by coordinate system. It is, therefore, suggested that use of word “space”
independent of coordinate system should be avoided and if used it must be kept in mind that it
represents volumetric extent of a specific coordinate system. The concept of space, if used
without caution, leads to an inaccurate understanding of the laws of nature.
3. Once the meanings of terms are clear, “the system of reference” or “frame of reference” or
“rigid body system” or “observer” or “coordinate system” may be used interchangeably to
denote an unique system for determination of motional quantities and the representation of a
Coordinate system types
Coordinate system types determine position of a point with measurements of distance or angle or
combination of them. A spatial point requires three measurements in each of these coordinate
types. It must, however, be noted that the descriptions of a point in any of these systems are
equivalent. Different coordinate types are mere convenience of appropriateness for a given
situation. Three major coordinate systems used in the study of physics are :
Rectangular (Cartesian) coordinate system is the most convenient as it is easy to visualize and
associate with our perception of motion in daily life. Spherical and cylindrical systems are
specifically designed to describe motions, which follow spherical or cylindrical curvatures.
Rectangular (Cartesian) coordinate system
The measurements of distances along three mutually perpendicular directions, designated as x,y
and z, completely define a point A (x,y,z).
Figure 1.10. A point in rectangular coordinate system
A point is specified with three coordinate values
From geometric consideration of triangle OAB,
From geometric consideration of triangle OBD,
Combining above two relations, we have :
The numbers are assigned to a point in the sequence x, y, z as shown for the points A and B.
Figure 1.11. Specifying points in rectangular coordinate system
A point is specified with coordinate values
Rectangular coordinate system can also be viewed as volume composed of three rectangular
surfaces. The three surfaces are designated as a pair of axial designations like “xy” plane. We may
infer that the “xy” plane is defined by two lines (x and y axes) at right angle. Thus, there are “xy”,
“yz” and “zx” rectangular planes that make up the space (volumetric extent) of the coordinate
system (See figure).
Figure 1.12. Planes in rectangular coordinate system
Three mutually perpendicular planes define domain of rectangular system
The motion need not be extended in all three directions, but may be limited to two or one
dimensions. A circular motion, for example, can be represented in any of the three planes,
whereby only two axes with an origin will be required to describe motion. A linear motion, on the
other hand, will require representation in one dimension only.
Spherical coordinate system
A three dimensional point “A” in spherical coordinate system is considered to be located on a
sphere of a radius “r”. The point lies on a particular cross section (or plane) containing origin of
the coordinate system. This cross section makes an angle “θ” from the “zx” plane (also known as
longitude angle). Once the plane is identified, the angle, φ, that the line joining origin O to the
point A, makes with “z” axis, uniquely defines the point A (r, θ, φ).
Figure 1.13. Spherical coordinate system
A point is specified with three coordinate values
It must be realized here that we need to designate three values r, θ and φ to uniquely define the
point A. If we do not specify θ, the point could then lie any of the infinite numbers of possible
cross section through the sphere like A'(See Figure below).
Figure 1.14. Spherical coordinate system
A point is specified with three coordinate values
From geometric consideration of spherical coordinate system :
These relations can be easily obtained, if we know to determine projection of a directional
quantity like position vector. For example, the projection of "r" in "xy" plane is "r sinφ". In turn, projection of "r sinφ" along x-axis is ""r sinφ cosθ". Hence,
In the similar fashion, we can determine other relations.
Cylindrical coordinate system
A three dimensional point “A” in cylindrical coordinate system is considered to be located on a
cylinder of a radius “r”. The point lies on a particular cross section (or plane) containing origin of
the coordinate system. This cross section makes an angle “θ” from the “zx” plane. Once the plane
is identified, the height, z, parallel to vertical axis “z” uniquely defines the point A(r, θ, z)
Figure 1.15. Cylindrical coordinate system
A point is specified with three coordinate values
Distance represents the magnitude of motion in terms of the "length" of the path, covered by an
object during its motion. The terms "distance" and "distance covered" are interchangeably used to
represent the same length along the path of motion and are considered equivalent terms. Initial
and final positions of the object are mere start and end points of measurement and are not
sufficient to determine distance. It must be understood that the distance is measured by the length
covered, which may not necessarily be along the straight line joining initial and final positions.
The path of the motion between two positions is an important consideration for determining
distance. One of the paths between two points is the shortest path, which may or may not be
followed during the motion.
Distance is the length of path followed during a motion.
Figure 1.16. Distance
Distance depends on the choice of path between two points
In the diagram shown above, s 1 , represents the shortest distance between points A and B.
s 2 ≥ s 1
The concept of distance is associated with the magnitude of movement of an object during the
motion. It does not matter if the object goes further away or suddenly moves in a different
direction or reverses its path. The magnitude of movement keeps adding up so long the object
moves. This notion of distance implies that distance is not linked with any directional attribute.
The distance is, thus, a scalar quantity of motion, which is cumulative in nature.
An object may even return to its original position over a period of time without any “net” change
in position; the distance, however, will not be zero. To understand this aspect of distance, let us
consider a point object that follows a circular path starting from point A and returns to the initial
position as shown in the figure above. Though, there is no change in the position over the period
of motion; but the object, in the meantime, covers a circular path, whose length is equal to its
perimeter i.e. 2πr.
Generally, we choose the symbol 's' to denote distance. A distance is also represented in the form
of “∆s” as the distance covered in a given time interval ∆t. The symbol “∆” pronounced as “del”
signifies the change in the quantity before which it appears.
Distance is a scalar quantity but with a special feature. It does not take negative value unlike some
other scalar quantities like “charge”, which can assume both positive and negative values. The
very fact that the distance keeps increasing regardless of the direction, implies that distance for a
body in motion is always positive. Mathematically : s > 0
Since distance is the measurement of length, its dimensional formula is [L] and its SI
measurement unit is “meter”.
Distance – time plot
Distance – time plot is a simple plot of two scalar quantities along two axes. However, the nature
of distance imposes certain restrictions, which characterize "distance - time" plot.
The nature of "distance – time" plot, with reference to its characteristics, is summarized here :
1. Distance is a positive scalar quantity. As such, "the distance – time" plot is a curve in the first
quadrant of the two dimensional plot.
2. As distance keeps increasing during a motion, the slope of the curve is always positive.
3. When the object undergoing motion stops, then the plot becomes straight line parallel to time
axis so that distance is constant as shown in the figure here.
Figure 1.17. Distance - time plot
One important implication of the positive slope of the "distance - time" plot is that the curve never
drops below a level at any moment of time. Besides, it must be noted that "distance - time" plot is
handy in determining "instantaneous speed", but we choose to conclude the discussion of "distance
- time" plot as these aspects are separately covered in subsequent module.
Example 1.1. Distance – time plot
Question : A ball falling from an height ‘h’ strikes the ground. The distance covered during
the fall at the end of each second is shown in the figure for the first 5 seconds. Draw distance –
time plot for the motion during this period. Also, discuss the nature of the curve.
Figure 1.18. Motion of a falling ball
Solution : We have experienced that a free falling object falls with increasing speed under the
influence of gravity. The distance covered in successive time intervals increases with time.
The magnitudes of distance covered in successive seconds given in the plot illustrate this
point. In the plot between distance and time as shown, the origin of the reference (coordinate
system) is chosen to coincide with initial point of the motion.
Figure 1.19. Distance – time plot
From the plot, it is clear that the ball covers more distance as it nears the ground. The
"distance- time" curve during fall is, thus, flatter near start point and steeper near earth
surface. Can you guess the nature of plot when a ball is thrown up against gravity?
A ball falling from an height ‘h’ strikes a hard horizontal surface with increasing speed. On each
rebound, the height reached by the ball is half of the height it fell from. Draw "distance – time"
plot for the motion covering two consecutive strikes, emphasizing the nature of curve (ignore
actual calculation). Also determine the total distance covered during the motion.
Here we first estimate the manner in which distance is covered under gravity as the ball falls or
The distance- time curve during fall is flatter near start point and steeper near earth surface. On
the other hand, we can estimate that the distance- time curve, during rise, is steeper near the earth
surface (covers more distance due to greater speed) and flatter as it reaches the maximum height,
when speed of the ball becomes zero.
The "distance – time" plot of the motion of the ball, showing the nature of curve during motion, is
Figure 1.19. Distance – time plot
The origin of plot (O) coincides with the initial position of the ball (t = 0). Before striking the
surface for the first time (A), it travels a distance of ‘h’. On rebound, it rises to a height of ‘h/2’
(B on plot). Total distance is ‘h + h/2 = 3h/2’. Again falling from a height of ‘h/2’, it strikes the
surface, covering a distance of ‘h/2’. The total distance from the start to the second strike (C on
plot) is ‘3h/2 + h/2 = 2h’.
Coordinate system enables us to specify a point in its defined volumetric space. We must
recognize that a point is a concept without dimensions; whereas the objects or bodies under
motion themselves are not points. The real bodies, however, approximates a point in translational