Get the detailed notes of motion in a straight line

1. MOTION IN A STRAIGHT LINE
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Motion and Rest
Distance and Displacement
Uniform Motion
Non-uniform Motion
Speed
Velocity
Acceleration
Equations of Uniformly Accelerated Motion
Graphical Representation of Motion
Distance-Time Graph
Speed-Time Graph
Derivation of Equations of Motion by Graphical Method
Uniform Circular Motion
Calculation of Speed of a Body in Uniform Circular Motion

2. Concept of a Point Object
In mechanics, a particle is a geometrical mass point or a material body of
negligible dimensions. It is only a mathematical idealization.
Examples:
Earth
In practice, the nearest approach to a particle is a body, whose size is much smaller
than the distance or the length measurements involved.

3. Rectilinear motion: The study of motion of objects along a straight line.
Ex. Freely falling body, motion of vehicle along straight road.
Mechanics: it is the branch of physics which deals with the motion and rest
objects.
Statics: study of
objects at rest
under the action
of force.
Dynamics: Study of objects in
motions under the consideration of
forces.
Kinematics: study of objects in motion
without considering the cause for the motion.
Ex. Freely falling body motion.
Kinetics: study of motion with
consideration of cause for the
motion.
Ex. Gravitation, friction.

4. Motion:
An object is said to be in motion if it changes its position with respect to its
surroundings and with time.
Examples:
1. Moving cars, buses, trains, cricket ball, etc.
2. All the planets revolving around the Sun.
3. Molecules of a gas in motion above 0 K.
Rest:
An object is said to be at rest if it does not change its position with respect to its
surroundings and with time.
Examples: Mountains, Buildings, trees etc.
Rest and Motion are relative terms
An object which is at rest can also be in motion simultaneously.
Eg. The passengers sitting in a moving train are at rest w.r.t. each other but they
are also in motion at the same time w.r.t. the objects like trees, buildings, etc.

5. Motions are classified based
on path into 3 types.
1D-motion: motion of a body
along a straight line.
Ex. Freely falling body, motion
of car along straight road.
2D-motion: motion of a body
in a plane or along curved
path.
Ex. Ant moves on a floor,
lizard moves on a wall.
3D-motion: motion of a body
in space.
Ex. Bird flying in the sky,
Aeroplane.
Motion and Rest are Relative Terms – How?
In the examples of motion of ball and car, man is considered to be
at rest (stationary).
But, the man is standing on the Earth and the Earth itself moves
around the Sun as well as rotates about its own axis.
Therefore, man is at rest w.r.t. the Earth but is rotating and
revolving around the Sun.
That is why motion and rest are relative terms !

6. Reference Point or Origin
While describing motion, we use reference point or origin w.r.t. which the motion of other bodies are
observed.
When you travel in a bus or train you can see
the trees, buildings and the poles moving back.
To a tree, you are moving forward and to you, the trees are moving back.
Both, you and the trees, can serve as reference point but motion can not be described without
reference point.
What effect do you get when you play video game involving car racing?
Frame of reference: The place from which motion is observed and measured is called F.R
Frame of reference very important for measurement of a body
motion.

7. Motion in a straight line: motion of a body along any one axis.
-x -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 +x
(in km)
1. The distance measured to the right of the origin of the position axis is taken positive
and the distance measured to the left of the origin is taken negative.
2. The origin for position can be shifted to any point on the position axis.
3. The distance between two points on position-axis is not affected due to the shift in
the origin of position-axis.
•
•
Path length or distance : It is the total distance covered by the body during motion.
It is a scalar and SI unit is Metre.
It gives always +ve value.

8. Displacement: it is the shortest path between initial and final position of the body during its motion.
Displacement = Final position –Initial position
Δx = x2 – x1 if x2 > x1, Δx is positive
x2 < x1, Δx is negative
if initial and final position of a body at the same point then displacement is zero.
X1
X2
Scalar
Scalar quantity is a physical quantity which has magnitude
only.
Eg.: Length, Mass, Time, Speed, Energy, etc.
Vector
Vector quantity is a physical quantity which has both
magnitude as well as direction.
Eg.: Displacement, Velocity, Acceleration, Momentum, Force,
etc.

9. Speed: it is defined as the ratio of the path length to the time taken.
speed SI unit is m/s.
1.
Types of speed:
Average speed: it is the ratio of the total path length covered by the body to the total time taken
=
X1+X2 ………….Xn
t1+t2 ………..….tn
2. Uniform speed: if a body covers equal distances in equal time intervals.
3.Non uniform speed or variable speed: if a body covers unequal distances in equal time intervals.
4.Instantaneous speed: limit of average speed when time interval is infinitesimally small.
(or) speed of the body at a particular instant of time.
Instantaneous speed = x derivative w.r.t time t

10. Velocity : it is defined as the ratio of the displacement to the time taken (or) rate of change of displacement.
velocity SI unit is m/s.
1.
Types of velocity:
Average velocity: it is the ratio of the total displacement covered by the body to the total time taken
=
X1+X2 ………….Xn
t1+t2 ………..….tn
2. Uniform velocity : if a body covers equal displacements in equal time intervals.
3.Non uniform velocity or variable velocity : if a body covers unequal displacements in equal time intervals.
4.Instantaneous velocity : limit of average velocity of a body, when time interval is infinitesimally small.
(or) velocity of the body at a particular instant of time.
Instantaneous velocity = x derivative w.r.t time t

11. 1.
Motions are two types based on the speed/velocity they are;
Uniform motion : in this motion body speed/velocity remains constant throughout the motion.
2. non uniform motion: in this motion body speed/velocity does not constant throughout the motion.

12. Difference between Speed and Velocity
Speed Velocity
1. Speed is the time rate of change of
distance of a body.
1. Velocity is the time rate of change
of displacement of a body.
2. Speed tells nothing about the
direction of motion of the body.
2. Velocity tells the direction of
motion of the body.
4. Speed of the body can be positive
or zero.
4. Velocity of the body can be
positive, zero or negative.
3. Speed is a scalar quantity. 3. Velocity is a vector quantity.
5. Average speed of amoving
body can never be zero.
5. Average velocity of a moving body
can be zero.

13. 1.
2.
Acceleration: the rate of change of velocity of a body in motion. SI unit is m/s2
A body is said to have acceleration
If there is change in the magnitude of velocity.
If there is change in the direction of velocity.
1.
Accelerations are four types they are
Uniform(constant) acceleration: if body velocity changes equal interval in equal interval of time.
(or) acceleration of body is constant in motion.
Ex. Acceleration due to gravity(freely falling body acceleration) = 9.8 m/s2
2. Non uniform(variable) acceleration: if body velocity changes unequal interval in equal interval of
time.
Ex. A vehicle moving on a crowded road.
3. Average acceleration: it is defined as change in velocity divided by the time interval.
Average acceleration(a) =
change in velocity
total time taken
=
t
v-u

14. 4. Instantaneous acceleration: it is the acceleration of the body at a instant of time, time interval is
infinitesimally small.
Graphical representation of motion of body:
Position-time graphs:
Note: slope of distance time graph gives speed
slope of displacement time graph gives
velocity

15. Velocity-time graphs:
Note: slope of v-t graph gives acceleration.
area of v-t graph gives displacement.
Note: velocity and speed remain same in uniform motion
along a straight line motion.

16. Acceleration-time graphs
a
a a

17. EQUATIONS OF UNIFORMLY ACCELERATED MOTION BY GRAPHICAL METHOD
First equation of motion:
Acceleration =
Time taken for change
Change in velocity
a =
BC
AC
a =
v - u
t
v – u = at
or v = u + at
Consider a body moving with initial velocity ‘u’ accelerates at uniform rate ‘a’. Let
‘v’ be the final velocity after time ‘t’ and ‘s’ be the displacement.
V
e
l
o
c
i
t
y
(
m
/
s
)
O D
B
v
u
E
A C
Time (s)
t
v-u
AC=OD=t
O tn
sn
Sn-1
Time (s)
tn-1
d
i
s
p
l
a
c
e
m
e
n
t
Body covers displacement between successive
time interval is given by
Sn - Sn-1=un+1/2(an2 ) – u(n-1) +1/2(a (n-1)2)
s = u + a(n – ½)
s = u + a(n – ½)
Freely fall body

18. Second equation of motion:
The area of trapezium OABC gives
the distance travelled.
s = ½ x AC x (OA + DB)
s = ½ x t x (u + v)
s = ½ x t x (u + u + at)
s = ½ x (2ut + at2)
s = ut + ½ at2
displacement = area of trapezium = ½(sum of two sides) height
Third equation of motion:
Displacement =average velocity x time
s = ½(u + v)t
= ½(u + v) v - u
a
2as =(v+u)(v-u)
= v2-u2
v2-u2
=2as
Note: average
Velocity v = ½(u+v)
V
e
l
o
c
i
t
y
(
m
/
s
)
Time (s)
O
C
B
v
u
E
A
D
t
v-u
v
t

19. Hw : Write a motion of equation for freely fall body and upward projected body under acceleration due to gravity.
a=+g for freely fall, a= -g for upward projected body.

20. Equations of motion by Calculus method
1. velocity-time relation.
Acceleration
Integrating on both sides
This is velocity-time relation.
velocity
Integrating on both sides
2.Position- time relation.

21. 3.Velocity-Position relation.
Integrating on both
sides
This is velocity-Position relation.
Note: we can use this calculus method for non
uniform acceleration body motion.

22. Consider two objects A and B moving
uniformly with average velocities vA and vB along x-axis. If xA (0) and xB (0) are positions of
objects A and B,respectively at time t = 0, their positions xA (t) and xB (t) at time t are given
by:
xA (t ) = xA (0) + vA t --------1
xB (t) = xB (0) + vB t --------2
displacement of object B w.r.t. object A is
xBA(t) = xB (t) – xA (t)
xBA(t) = [ xB (0) – xA (0) ] + (vB – vA) t. ----3
Object B has a velocity vB – vA w.r.t. A
Relative velocity: it is the velocity of one body w.r.t the velocity of another body.
The velocity of object B relative to object A is vB – vA
vBA = vB – vA
The velocity of object A relative to object B is vA –
vB
vAB = vA – vB
At meet point XAB(t) is zero;
then meeting time is given by
t = [ xA (0) – xB (0) ] / vA-vB vA>vB

23. O
2
0
4
0
6
0
80
10
0
12
0
14
0
X(m)
t(s)
O
20
40
60
80
100
120
140
t(s)
X(m)
1 2 3 4 5 6
1 2 3 4 5 6
A
B
A
B
vA =
vB
vA – vB =
0
vB - vA = 0
The two objects stay
at a constant distance
vA >
vB
vB - vA is
negative
Objects meet at
a
common point

24. O
2
0
4
0
6
0
8
0
10
0
12
0
14
0
t(s)
X(m)
1 2 3 4 5 6
A
B
vA and vB are of opposite
signs.
In this case, the magnitude of vBA or vAB is
greater than the magnitude of velocity of A
or
that of B