Motion (1)

What Is Motion
Motion is the change in position of a body with
time. Motion can be described in terms of :-
1.Distance Travelled
2. Displacement

Distance travelled and Displacement
• When a body moves from one point to another, the distance
travelled refers to the distance travelled refers to the actual
length of the path travelled by a body.
• When, a body moves from one position to another, the
shortest distance between the initial position and final
position of the body, alongwith direction is known as its
displacement.
displacement =
• Example:- If a body starts moving in a straight line from origin O
and moves through C and B and reaches A and then moves back
and reaches C through B, then
Distance travelled =70 km+40 km=110 km
Displacement =70 km-40 km= 30 km
O B A
70 km
30km 20 kmC 20 km

The distance travelled by an object cannot be zero but the final displacement of a
moving body can be zero. The displacement of a moving object can be zero if, after
travelling a certain distance, the moving body finally comes back to its starting point.
This will become clear from the following examples.
Example 1. Suppose a man starts from place A and travels a distance of 5km to reach
place B. From place B he travels 3km and reaches C. And finally the man travels 4km
from C to reach the starting point A. In this case, though the man has travelled a
distance of 12 km, but the final displacement of the man is 0. This is because the man
has reached back at the starting point A and the straight line between initial A and final
position A is 0.
Example 2. If we travel along a circular along a radius r and reach back at the starting
point A , then though we have travelled the distance of 2πr but our final displacement
is 0.
3km
4km
r
A

Uniform and Non-uniform Motion
Uniform motion
A body has a uniform motion if it travels equal distances in equal
intervals of time, no matter how small these time intervals may be. For
e.g., a car running at a constant speed of say,10 meters per second, will
o e e ual dista es of ete s, e e y se o d ,so it s otio ill e
uniform.
Non-uniform motion
A body has a non-uniform motion if it travels unequal distance in
unequal intervals of time. For e.g. ,if we drop a ball from the roof of a
tall building, we will find that it in non uniform motion. It covers,
4.9 meters in the 1st second
14.7 meters in the 2nd second, and so on

Speed
Speed of a body gives us an idea of how slow or how fast a thing
is moving. We can define the speed of a thing as follows: Speed of
a body is the distance travelled by it per unit time. The speed of
the object can be written as:
=s
where =speed
s=distance travelled
t=time taken
S.I. unit of speed is m/s
t

Average Speed
The average speed of a body is the total speed travelled divided by
the total time taken to cover the distance. For example, a car
which travels a distance of 100km in 4 hrs, the average speed is
100km/4h s = 5k /h , ut it does t ea that a is o i g at
this speed all the time. When the road is straight, flat and free,
the speed may be much more than 25km/hr but on curves, hills or
in a crowded area, the speed may fall below this average value.
Formula for average speed is:
Average speed=
Uniform Speed
A body has a uniform speed if it travels equal distances in equal
intervals of time
Total distance travelled
Total Time taken

Velocity
The speed of a car gives us an idea of how fast the car is moving but it
does not tell us about the direction in which the car is moving. Thus to
know the exact position of a car we should also know the direction in
which the car is moving. In other words, we should know the speed of
car as well as the direction of car. This gives us another term as velocity,
which can be defined as follows: Velocity of a body is the distance
travelled by it per unit time in a given direction. We know that the
dista e t a elled i a gi e di e tio is k o as displa e e t . So
we can also write the definition in terms of displacement. We can say
that: Velocity is displacement per unit time. That is
Velocity =
S.I. Unit of velocity is same as that of speed, namely, m/s.
displacement
time taken

Average Velocity
In case the velocity of the object is changing at a uniform rate, then
average velocity is given by the arithmetic mean of initial velocity is
known and final velocity for a given period of time. That is,
=
where =velocity
u= initial velocity
v=final velocity
Uniform Velocity
A body has a uniform velocity if it travels in a specified direction in
a straight line and moves over equal distances in equal intervals of
time.
u + 
2

Acceleration
When the velocity of a body is increasing, the body is said to be
accelerating. Suppose a car starts off from rest and its velocity
increases at a steady rate so that after 5 sec velocity is 10m/s.
Now, in 5 sec the velocity has increased by 10-0 = 10m/s and in
one second the velocity increases by 10/5 = 2m/s. In other words,
the rate at which the velocity is increases is 2m/s in every second.
The car is said to have an acceleration of 2m/s per second. This
gives us the definition of acceleration, that is: Acceleration of a
object is defined as the rate of change of its velocity with time.
That is ,
Acceleration =
S.I. unit of acceleration is m/s 2
Time taken for change
Change in velocity

Uniform Acceleration
When the velocity of a car increases, the car is said to be in accelerating. If the
velocity increases at a uniform rate, the acceleration is said to be uniform . A
body has a uniform acceleration if it travels in a straight line if its velocity
increases by equal amounts in equal intervals of time. Here are some
examples of uniformly accelerated motion:
1.The motion of a freely falling ball is uniformly accelerated motion.
2.The motion of a ball rolling down in an inclined plane is an example of
uniformly accelerated motion.
Non-Uniform Acceleration
A body has anon uniform acceleration if its velocity increases by unequal
amount in equal intervals of time. In other words, a body has uniform a
acceleration if its velocity changes at uniform rate. The velocity if a car
running in a crowded city road changes continuously. At one moment the
velocity of a car increases where as at another moment it decreases. So
example of a car in a crowded city is an example of non uniform acceleration.

Retardation or Deceleration
Acceleration takes place when the velocity of a body changes. The velocity
body may increase or decrease, accordingly the acceleration is of two types-
positive acceleration and negative acceleration. If the velocity of an object
increases it is called acceleration and if the velocity of an object decreases it is
called Retardation or deceleration. Retardation is measured in the same way as
acceleration, that is equal to and has the same units of
m/s2. Here is one e.g. When a car driver travelling at an initial velocity of 10m/s
applies brakes and brings the car to rest in 5 seconds, then:
Acceleration ,a =
a=
a= -2m/s2
Thus, the acceleration of the car is, -2m/s2. It is negative in sign , but the
negative acceleration is known as retardation, so the car has a retardation of
2m/s2.
Change in velocity
Final velocity – Initial velocity
Time Taken
0m/s-10m/s
5

Graphical Representation of Motion
Distance Time Graph
The change in the position of a body with time can be represented on the
distance time graph. In this graph distance is taken on the y – axis and time is
taken on the x – axis. The slope of a distance time graph indicates speed of
the body.
1. The distance time graph for uniform speed is a straight line ( linear ).
This is because in uniform speed a body travels equal distances in equal
intervals of time. We can determine the speed of the body from the distance
– time graph. For the speed of the body between the points A and B, distance
is (s 2 – s1) and time is (t2 – t1).
=
=
=
= 2m/s
s 2- t2
t2 – t1
20-10
10-5
10
5 10
20
5 10
0
distance
time
Y
X

2.The distance – time graph for non uniform motion is non
linear. This is because in non uniform speed a body travels
unequal distances in equal intervals of time.
0
10
15
25
5
30
20
Time (s)
Distance(m)
Y
X
35
40
45
2 4 6 8 10

Velocity – time graphs
The change in the velocity of a body with time can be represented
on the velocity time graph. In this graph velocity is taken on the y –
axis and time is taken on the x – axis.
2. If a body moves with uniform velocity, the graph will be a
straight line parallel to the x – axis . This is because the velocity
does not change with time. To determine the distance travelled by
the body between the points A and B with velocity 20 km/h.
=
s= *t
= 20 km/h= AC or BD
t = t2 – t1 = DC = AC (t2 – t1)
s = AC*CD
s = area of the rectangle ABDC
s
t
10
20
30
40
5 1510 20
0
Velocity(km/h)
Time (s)
A B
C D
X
Y

ii) If a body whose velocity is increasing with time, the graph is a
straight line having an increasing slope. This is because the
velocity increases by equal amounts with equal intervals of time.
The area under the velocity – time graph is the distance
(magnitude of displacement) of the body. The distance travelled
by a body between the points A and E is the area ABCDE under
the velocity – time graph.
s = area ABCDE
= area of rectangle ABCD
+ area of triangle ADE
s = AB*BC+ (AC * DE)
1
2
10
20
30
40
0
10 20 30 40
A
B C
D
E

iii) If a body whose velocity is decreasing with time, the graph is a straight
line having an decreasing slope. This is because the velocity decreases
by equal amounts with equal intervals of time.
iv) If a body whose velocity is non uniform, the graph shows different
variations. This is because the velocity changes by unequal amounts in
equal intervals of time. 20 40 Time (s) Velocity(ms-1 ) X 10 30 50 10 15
20 20 40 Time (s) Velocity(ms-1 ) X 10 30 50 10 15 20 Velocity – time
graph for a uniformly decelerated motion Velocity – time graph for non
uniform acceleration Y .
Velocity(m/s)
Time (s)
Velocity(m/s)
Time (s)

Equations of Motion by Graphical
Representation
When an object moves along a straight line in a uniform acceleration, it is
possible to relate its velocity, acceleration and the distance covered by it
in a certain interval by a set of equation known as the equations of
motion. There are three such equations. These are:
 =u + at
s=ut + ½ at 2
2as=  2 –u 2
Where u is the initial velocity of the object which moves with uniform
acceleration a for time t ,  is the final velocity, and s is the distance
travelled by the object in time t .The first equation describes the velocity-
time relation and second equation represents the position-time relation.
The third equation, which represents the relation between the position
and the velocity, can be obtained form first and second equations by
eliminating t. These three equations can be deprived by graphical
representation.

Equation for Velocity Time Relation
Consider a velocity – ti e g aph fo a ody o i g ith u ifo a ele atio a .
The initial velocity is u at A and final velocity is v at B in time t.
Perpendicular lines BC and BE are drawn from point B to the time and velocity axes
so that the initial velocity is OA and final velocity is BC and time interval is OC.
Draw AD parallel to OC.
We observe that
BC = BD + DC = BD + OA
Substituting BC =  and OA = u
We Get  = BD + u
or BD =  - u _ _ _ _ _ _ _ _ _ _ _ (1)
From the velocity –time graph the acceleration of the object is given by
a =
= =
Substituting OC we get t
a=
or BD=at _ _ _ _ _ _ _ _ _ _ _ _ (2)
Using Eqs. (1) and (2) we get  = u + at
Change in velocity
BD
AD
BD
OC
BD
t A
B
C
D
E
O
t
v
u
Velocity(m/s)
Time (s)

Equation for Position- Time Relation
Consider a velocity – time graph for a body moving with uniform
a ele atio a t a elled a dista e s in time t.
The distance traveled by the body between the points A and B is
the area OABC.
s = area OABC ( which is a trapezium )
= area of rectangle OABC + area of triangle ABD
= OA X OC + ½ (AD * DE )
Substituting OA = u, OC = AD = t,
BD = v – u = at
We get
s = ut + ½ at 2

Equation for Position-velocity Relation
Consider a velocity – time graph for a body moving with uniform
a ele atio a t a elled a dista e s i ti e t. The dista e
travelled by the body between the points A and B is the area
OABC.
=
Substituting OA = u, BC =  and OC = t, we get,
s = _ _ _ _ _ _ _ _ _ _ _ _ (1)
From the velocity-time relation Eq. (1), we get
t = _ _ _ _ _ _ _ _ _ _ _ _ _ (2)
Using Equations (1) and (2) we have
s =
or 2as=  2 –u 2
(0A + BC ) * OC
2
(u + )t
2
( - u )
a
( + u ) * (  - u )
2a
s

Circular Motion
The motion of a body moving around a fixed point in the circular
path is known as circular motion.
Uniform Circular Motion
When a is body moving in a circular motion with a uniform speed
it is said to be in uniform circular motion.
Direction of motion of a body moving in a circular path at any
instant is along a tangent to the position of body, On circular path
at that instant.
In a circular motion the velocity of an object is given by where
2πr is the distance covered by an object.
2πr
t
The velocity is constant, but the velocity
is always tangent to the orbit; the
acceleration has constant magnitude,
but always points toward the center of
rotation.
velocity
Acceleration

Motion (1)

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Motion (1)