SCALAR & VECTOR QUANTITIES
REST & MOTION
SCALAR
VECTOR
Physical quantities that
are only expressed by
their magnitude.
Physical quantities that are
expressed
with
both
magnitude and direction.
Scalars
have
two
components
or
parameters – Numeric
value of the quantity
(magnitude) and unit of
measurement
Vectors
have
3
components–
Numeric
value of the quantity
(magnitude),
unit
of
measurement
and
direction
Scalars are non-zero
positive real numbers
that can be added,
subtracted, multiplied,
and divided by simple
arithmetic methods
Vectors can be integer
value, where the negative
sign denotes reverse or
opposite direction. They
follow vector algebra for
their arithmetic operations
They are denoted by a
symbol or letter of the
alphabet.
Vectors are denoted by
symbol or letter of the
alphabet along with an
arrow on it. E.g., 𝐹 ⃗
E.g., mass, length, time,
distance,
density,
volume, speed, energy
etc
E.g., displacement, weight,
force, velocity, torque etc
CH2_CS02_MOTION IN ONE DIMENSION
DISTANCE & DISPLACEMENT
• A body is said to be at rest if it does not change its
position with respect to its immediate surroundings
over time.
• A body is said to be in motion if it changes its position
with respect to its immediate surroundings over time.
ANALYSIS OF MOTION
• When a body moves along a straight line, its motion is
said to be one dimensional motion. In such
a motion, there is no lateral/sideways
motion.
• When a body moves along a curved path,
its motion is two-dimensional while its
motion in space is three-dimensional.
REPRESENTATION OF ONE-DIMENSIONAL MOTION
One-dimensional motion can be represented by a straight
line parallel to the X-axis in the direction of motion. Each
point on the straight line represents the position of the
particle in different time instants. The position of particle
at any time instant - ‘t’ is expressed by specifying the xcoordinate at that instant of time. As the particle moves,
its x-coordinate will change with time ‘t’.
DISTANCE (S)
⃗⃗)
DISPLACEMENT (𝑺
Length of the actual
path traversed by the
body
Shortest distance between the
initial and final positions of the
body
Scalar
Vector
Depends on the path
taken
Does not depend on the path
taken
Positive quantity
Can be positive, negative or
zero
Magnitude of distance
is equal to or more than
the displacement
Magnitude of displacement is
equal to or less than the
distance
May not be zero even if
displacement is zero. Is
never
zero
if
displacement is nonzero
Can be zero even when
distance is non-zero
NOTE: Distance and displacement of a moving body is
the same when the body is moving along a straight-line
path, in a given direction.
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SPEED AND VELOCITY
DIFFERENT TYPES OF SPEED
SPEED (v)
⃗⃗)
VELOCITY (𝒗
Distance travelled per
second by a moving
body
Displacement per second of
a moving body
Scalar
Vector
Positive quantity
Can be positive or negative
After one round in a
circular path, average
speed is non-zero
After one round in a circular
path, average velocity is
zero
CH2_CS02_MOTION IN ONE DIMENSION
• Instantaneous velocity: Ratio of the displacement of
the body in a small-time interval to the time interval.
• Average velocity: Ratio of the total displacement to
the total time taken for the entire journey.
• A body is said to be moving with uniform speed if it
covers equal distance in equal intervals of time
throughout its motion. E.g., Motion of a ball on a
frictionless plane surface.
• A body is said to be moving with non-uniform or
BODY WITH CONSTANT SPEED, VARIABLE VELOCITY
variable speed if it covers unequal distances in equal
intervals of time. E.g., motion of a car through a
crowded street
• Instantaneous speed is the speed at any instant of time.
It is equal to the ratio of the distance travelled in a short
time interval to the time interval. E.g., speed value
noted in vehicle’s speedometer
• Average speed is the ratio of the total distance travelled
by the total time taken to travel that distance.
When a particle moves with a constant speed in a
NOTE: For a body travelling with uniform speed, the
circular path, its motion is said to be uniform circular
instantaneous speed is equal to the average speed.
motion.
DIFFERENT TYPES OF VELOCITY
Speed is a constant in such a motion; however velocity
• Uniform velocity: When body travels equal distances, in is variable as the direction of the body keeps changing.
a particular direction, in equal intervals of time. E.g. rain The direction of motion of the particle is at any point is
drops reach earth’s surface with uniform velocity.
given by the tangent drawn at that point of the circular
• Non-uniform or Variable velocity: When body travels path.
unequal distances, in a particular direction, in equal
intervals of time or when the body moves equal Unlike uniform linear motion, where speed & velocity
NOTE: Speed and velocity of a moving body are the
distances in equal intervals of time, but its direction of both are constant, and acceleration is zero; uniform
same when the distance covered by it and its
motion is not the same. E.g., Motion of a free-falling circular motion is accelerated as the velocity keeps
displacement are the same, i.e., it is moving along a
changing.
object.
straight-line path.
Page 2 of 7
CH2_CS02_MOTION IN ONE DIMENSION
Graph Implications
ACCELERATION: Acceleration (𝑎⃗) is defined as the ACCELERATION DUE TO GRAVITY
rate of change of velocity with time; 𝑎⃗ =
⃗⃗
𝑣
𝑡
Acceleration due to gravity (g): when a body falls freely POSITIVE SLOPE: body is moving away from the starting
under gravity, the acceleration produced in the body due or reference point
S.I. unit: m/s2 or m s-2
CGS unit: cm/s2 or cm s-2
to the earth’s gravitational acceleration.
NEGATIVE SLOPE: body is returning towards the
When velocity changes from initial velocity (𝑢
⃗⃗) to
This does not depend on the mass of the body. Thus, two starting or reference point
final velocity (𝑣⃗), acceleration (𝑎⃗) is expressed as –
bodies of different masses when simultaneously dropped
EXAMPLES
𝑣⃗ − 𝑢
⃗⃗
in vacuum will reach ground at the same time.
𝑎⃗ =
𝑡
1. Body at rest / Stationary object
Negative acceleration or deceleration or retardation;
is the acceleration of a body that is slowing down GRAPHICAL REPRESENTATION OF LINEAR MOTION
(when 𝑢
⃗⃗ > 𝑣⃗)
If a body moves in a straight-line path, its motion is in one
dimension and is called linear or rectilinear motion. A
linear motion can be studied using the below graphs –
• Displacement time graph
• Velocity time graph
•
•
•
• Acceleration time graph
TYPES OF ACCELERATION
• Uniform acceleration: when equal changes in
velocity takes place in equal intervals of time.
• Variable acceleration: when changes in velocity is
not the same in equal intervals of time.
DISPLACEMENT-TIME
GRAPHS (x-t graphs)
GRAPHS
OR
POSITION-TIME
In this graph, time is plotted along the X-axis while
displacement is along the Y-axis. We can use this graph to
determine the velocity of the body. Since velocity is the
ratio of displacement and time, the slope of the
displacement time graph gives the velocity.
•
•
•
Displacement from origin remains constant
Graph is a straight line parallel to the X-axis
Body is at rest or stationary at a certain
distance from the origin / reference point.
OA is the distance from the reference point or
origin. Distance OA remains constant at all
instants of time.
For an object at origin itself, the graph will be
coinciding with the time axis.
Slope of the graph is zero. Hence velocity is also
zero.
Page 3 of 7
2. Body moving
uniform velocity
with
• Displacement increases by
the same amount in each
time interval/each second
• Graph is a straight line inclined to the time axis.
• Velocity of the body can be obtained by finding the
slope of the straight line.
• Greater the slope (i.e., angle made with the time
axis), higher is the velocity,
• For a body moving with uniform velocity, the
displacement is directly proportional to the time
taken (s  t).
3. Body with variable velocity
VELOCITY – TIME GRAPHS
In the velocity time graphs, time is taken on the X-axis
while velocity is taken on the Y-axis. From the velocity
time graph, we can determine the following –
• Acceleration of the body at any instant: to determine the
acceleration of the body at a given instant of time, we take
the slope of the velocity time graph.
• Displacement of the body in a certain time interval: To
determine the displacement in a certain time interval
from a velocity time graph, the area enclosed between the
velocity-time sketch and the X-axis is calculated.
EXAMPLES
1. Body moving with uniform velocity
• Displacement
time
graph for a body
moving with nonuniform velocity is a
curve
• Velocity can be obtained by finding the slope /
gradient of the tangent drawn on the curve at that
instant of time. Slope will be different at different
points of the graph
• Velocity varies at different instants of time
NOTE: X-T graph can never be a straight line parallel
to the Y-axis as this would imply that the distance
covered by the body in a certain direction increases
without any increase in time, which is impossible.
CH2_CS02_MOTION IN ONE DIMENSION
2. Body moving with uniform acceleration
• Body has equal changes in velocity in equal intervals
of time.
• Larger the slope, greater is the acceleration
• Graph is a straight line inclined to the X-axis
• Slope gives acceleration
• Area enclosed by curve and X-axis will give the
displacement.
3. Body moving with uniform retardation
•
•
•
•
velocity remains constant at all times
Acceleration is zero
Graph is parallel to the X-axis
Slope AB is zero
• Velocity decreases by an equal amount in equal
intervals of time.
• Velocity time graph is a straight line inclined to the Xaxis with a negative slope. Larger the slope, greater is
the retardation.
Page 4 of 7
4. Body initially not at rest, but later moving with ACCELERATION-TIME GRAPHS: Here acceleration is
uniform velocity
plotted along the Y-axis while time is taken on the X-axis.
From this graph, the change in speed in a certain interval
of time can be determined, from the area enclosed
between the acceleration time graph and the X-axis.
• Graph is a straight line inclined at zero
• Does not begin at origin.
Distance & Displacement Calculation Using V-T
Graphs
1. Body moving with uniform velocity: For a stationery
object or a body moving with
uniform velocity, the acceleration is
zero. Thus, the A-T graph is a
straight-line coinciding with the time
axis (X-axis).
CH2_CS02_MOTION IN ONE DIMENSION
MOTION UNDER GRAVITY: A body falling freely under
gravity moves with uniform acceleration of 9.8 m/s2.
For a body rising vertically upwards, there is a uniform
retardation of 9.8 m/s2.
Here we are approximating the value to be 10 m/s2
Acceleration-Time Graph of Free-falling Body
2. Body moving with uniform acceleration: For a body
moving with uniform acceleration,
the velocity increases uniformly in
equal intervals of time. Thus, the
graph is a straight line parallel to the
time axis on the positive side of the
acceleration axis.
Obtain the V-T graph from the above, by finding the
area enclosed by the straight line and the time axis for
each time interval of 1 s. Let initial velocity at t = 0 be
3. Body moving with uniform retardation: For a body zero. Then
moving with uniform retardation, the
Velocity after 1 s = Area OABP = 10 x 1 = 10 m/s
velocity of the body decreases
uniformly in equal intervals of time.
Velocity after 2 s = Area OACQ = 10 x 2 = 20 m/s
Thus, the graph is a straight line
Velocity after 3 s = Area OADR = 10 x 3 = 30 m/s
For the given graph, the vector sum of the area of the parallel to the time axis, on the
trapezium and area of triangle would give the negative side of the acceleration axis.
And so on…
displacement of the body. Area above the X-axis
Time (s)
0
1
2
3
4
5
gives the positive displacement. Area below the Xaxis gives the negative displacement. In this example, NOTE: For a body moving with variable
Velocity
0
10
20
30
40
50
area of the trapezium + (- area of triangle) will give acceleration/retardation, graph will be a curve in any
(m/s)
the net displacement of the body.
shape
Page 5 of 7
V-T Graph of Free-Falling Body
CH2_CS02_MOTION IN ONE DIMENSION
EQUATIONS OF LINEAR MOTION: For motion of a body
moving with uniform acceleration, the following three
equations give the relationship between the following
X-T Graph of Free-Falling Body
•
•
•
•
•
initial velocity (𝑢
⃗⃗)
final velocity (𝑣⃗)
acceleration (𝑎⃗)
time of journey (t)
distance travelled (S)
NOTE: For a free-falling body, the displacement of the
Use the graph to obtain the displacement-time graph body is directly proportional to the square of the time (s
by finding the area enclosed by the straight line OA  t 2).
with the time-axis for each time interval of 1 s.
Square of
0
1
4
9
16
25
2
Displacement in 1 s = Area oap = ½ x 1 x 10 = 5 m
Time (s )
The equations are listed below –
Dispacement in 2 s = Area obq = ½ x 2 x 20 = 20 m
DERIVATION USING GRAPHICAL METHOD:
Displacement in 3 s = Area ocr = ½ x 3 x 30 = 45 m
Displacement in 4 s = Area ocr = ½ x 4 x 40 = 80 m
Displacement 0
(m)
5
20
45
80
125
(i)
(ii)
(iii)
v = u + at
s = ½ (u+v)t = ut + ½ at2
v2 = u2 + 2aS
For a body moving with uniform acceleration, the V-T
graph is a straight-line AB as shown. The initial velocity
of the body is u and final velocity is v, in time t.
Displacement in 5 s = Area ocr = ½ x 5 x 50 = 125 m
Time (s)
0
1
2
3
4
5
Position
(m)
0
5
20
45
80
125
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CH2_CS02_MOTION IN ONE DIMENSION
Page 7 of 7