This document covers the principles of motion in a straight line, a fundamental topic in mechanics, which is essential to physics. It discusses kinematics, including types of motion, displacement, velocity, acceleration, the equations of motion, and concepts like free fall and relative velocity. Key definitions and graphical interpretations are provided to illustrate motion concepts and formulas.

1. Motion in Straight Line
Chapter - 3

2.  Mechanics is the science that deals with motion of objects.
 It is basic to all other branches of physics.
 The branch of mechanics that describes the motion of objects without considering force
(cause of motion) is called kinematics.
 The moving object of concern is either a particle (a point-like object) or an object that
can be viewed to move like a particle.
Mechanics
In this branch we answer questions like “Does the object speed up, slow down, stop, or
reverse direction?” and “How is time involved in these situations?”

3. Motion
 Motion as the change of an object’s position with time.
 The path along which the object moves, whether straight or curved, is called the
object’s trajectory.
Four basic types of Motion
Linear motion Circular motion Projectile motion Rotational motion

4. In this chapter, we only study motion along
straight line called rectilinear motion.

5. Motion Diagram
Object is at rest.
A stationary ball on the ground
Object moving with constant
speed.
A skateboarder rolling the
sidewalk
Object is speeding up.
A sprinter starting the 100 meter dash
Object is slowing down.
A car stopping for a red light
Both slowing down (as the ball rises) and
speeding up (as the ball falls).
A jump shot from centre court

6. Idealized Model
In physics, a model is a simplified version of a physical system that would be too complicated to analyze in
full without the simplifications.
Real Physical System Idealized Model

7. Position and Displacement
The object’s position is its location with respect to a chosen reference point (origin of an axis). The
positive/negative direction of this axis is the direction of increasing/decreasing numbers.
Frame of Reference
A change in the object’s position from an initial position to a final position is called displacement
Displacement is a vector quantity which has a magnitude and a direction. Magnitude is the distance
between the initial and final positions and the direction is plus or minus for motion to the right or to the
left, respectively.
if is positive
if is negative
𝚫 𝒙=𝒙𝒇 −𝒙𝒊
if

8. Displacement
Displacement,
But actual length travelled by the object
Path length,
Position – Time
graph
Position t (s) x (m)
A 0 1
B 1 3
C 2 4
D 3 2
E 4 -1
A B
C
D
E
A
B
C
D
E

9. Position-Time Graph
𝑥
𝑡
𝑥
𝑡
𝑥
𝑥
𝑡
rest uniform motion speeding down
speeding up
𝑥 speeding down
in direction
𝑥
𝑡
in real situation

10. Average Velocity and Average Speed
The average velocity, , of a particle is defined as the ratio of its
displacement, , to the time interval, . That is:
𝒗𝒂𝒗𝒈=
𝜟 𝒙
𝜟 𝒕
=
𝒙 𝒇 −𝒙𝒊
𝒕𝒇 −𝒕𝒊
The average distance of a particle is defined as the ratio of its total path length to the total time interval.
That is:
𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑=
𝑇𝑜𝑡𝑎𝑙 h
𝑝𝑎𝑡 h
𝑙𝑒𝑛𝑔𝑡
𝑇𝑜𝑡𝑎𝑙𝑡𝑖𝑚𝑒𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
[Unit is m/s]
The average velocity is rate at which
the displacement occurs.

11. The average speed is not the magnitude of the average velocity.
• For example, a runner ends at her starting point.
• Her displacement is zero.
• Therefore, her velocity is zero.
• However, the distance traveled is not zero, so the speed is not zero.

12. Instantaneous Velocity and Speed
The instantaneous velocity of an object indicates how fast the object moves and the direction of the
motion at each instant of time. The magnitude of the instantaneous velocity is called the
instantaneous speed, and it is the number (with units).
This means that we evaluate the average velocity over a shorter and shorter period of time; as that time
becomes infinitesimally small, we have the instantaneous velocity.
𝒗=𝐥𝐢𝐦
𝜟 𝒕 →𝟎
𝜟 𝒙
𝜟𝒕
=
𝒅 𝒙
𝒅𝒕
This plot shows the average velocity being
measured over shorter and shorter intervals. The
instantaneous velocity is tangent to the curve.
Instantaneous velocity is simply called velocity

13. Instantaneous Velocity
Graphical Interpretation of Average
and Instantaneous Velocity
For uniform motion, velocity is the same as the average velocity of all instants.
𝑥
𝑡
𝑥
𝑡
𝑥
𝑡
𝒗=𝟎 is positive is negative

14. Acceleration
The average acceleration, , for a particular time interval is defined as the change in velocity, , divided
by that time interval. That is
𝒂𝒂𝒗𝒈 =
𝜟 𝒗
𝜟𝒕
=
𝒗𝒇 −𝒗𝒊
𝒕𝒇 −𝒕𝒊
Instantaneous acceleration is the limit of the ratio as approaches zero. On a plot of velocity versus
time, the instantaneous acceleration at time t is the slope of the line tangent to the curve at that time:
𝒂=𝐥𝐢𝐦
𝜟𝒕→𝟎
𝜟𝒗
𝜟𝒕
=
𝒅𝒗
𝒅𝒕
=¿
𝒅
𝒅𝒕 (𝒅 𝒙
𝒅𝒕 )=
𝒅𝟐
𝒙
𝒅𝒕
𝟐
¿
Instantaneous acceleration is simply called acceleration
[Unit is m/s2
]

15. Acceleration
Graphical Interpretation of Average and Instantaneous Acceleration:

19. • axis the second derivative of x with respect to t.
• The second derivative of any function is directly related to the
concavity or curvature of the graph of that function .
• Where the x-t graph is concave up (curved upward), the x-
acceleration is positive and is increasing; at a point where the x-t
graph is concave down (curved downward), the x-acceleration is
negative and is decreasing.
• At a point where the x-t graph has no curvature, such as an
inflection point, the x-acceleration is zero and the velocity is not
changing. Figure above shows all three of these possibilities.
Note:
Examining the curvature of an x-t graph is an easy way to decide
what the
sign of acceleration is. This technique is less helpful for determining
numerical

20. Constant Acceleration

21. Graph for Constant Acceleration
𝑣
𝑡
𝑣
𝑡
𝑣
𝑡
is constant is positive
is negative
𝒗 𝟎
𝒗 𝟎
𝒗 𝟎
𝑣
𝑡
is negative
𝒗 𝟎
𝑣
is negative
𝒗 𝟎
𝑣
𝑡
is positive
𝒗 𝟎
is the velocity at
𝑡

23. Kinematic Equations for Constant
Acceleration
𝑣 =𝑣0 +𝑎𝑡 (1)
For convenience, we let and , where t is any arbitrary time. Also, we let (the initial velocity at time ) and
(the velocity at any time t). With this notation, we can express acceleration as:
𝑎=
𝑑𝑣
𝑑𝑡
=
𝑣− 𝑣0
𝑡 − 0
=
𝑣− 𝑣0
𝑡
𝑎𝑡=𝑣 −𝑣0
As the acceleration is constant, the average velocity in any time interval is the arithmetic mean of the
initial velocity, , and the final velocity at the end of that interval, . Thus:
𝑣𝑎𝑣𝑔=
𝑣0 +𝑣
2
=
𝑥
𝑡
𝑥=𝑣0 𝑡+
1
2
𝑎𝑡
2
(3)
𝑥=
1
2
(𝑣0+ 𝑣 )𝑡(2) 𝑥=
1
2
(𝑣0+𝑎𝑡+𝑣0 )𝑡
𝑥=
𝑣 +𝑣0
2
𝑡=(𝑣 +𝑣0
2 )(𝑣 − 𝑣0
𝑎 )=
𝑣2
−𝑣0
2
2 𝑎
𝑣2
=𝑣0
2
+2 𝑎𝑥(4)
All three equations of are obtained by taking initial position at origin. If initial position of object is non-
zero, then initial position is taken as . Then equations (2), (3) and (4) become
𝑥=𝑥0+𝑣0 𝑡+
1
2
𝑎𝑡
2
(6) 𝑣2
=𝑣0
2
+2 𝑎( 𝑥 − 𝑥0 )(7)
𝑥=𝑥0+
1
2
(𝑣0 +𝑣 )𝑡 (5)

24. Equations of Motion with Constant
Acceleration
𝑣=𝑣0+𝑎𝑡
𝑣2
=𝑣0
2
+2 𝑎(𝑥 − 𝑥0 )
𝑥=𝑥0+𝑣0 𝑡+
1
2
𝑎𝑡2
𝑥=𝑥0+
1
2
(𝑣0 +𝑣 )𝑡
Equations Missing quantity
𝑥 − 𝑥0
𝑎
𝑣
𝑡

25. Equations of Motion with Constant
Acceleration
By Integral Calculus
𝑎=
𝑑𝑣
𝑑𝑡
𝑑𝑣=𝑎𝑑𝑡 ∫
𝑣0
𝑣
𝑑𝑣=¿∫
𝑡0
𝑡
𝑎𝑑𝑡¿ 𝑣 −𝑣0=𝑎𝑡 𝒗=𝒗𝟎 +𝒂𝒕
𝑣=
𝑑 𝑥
𝑑𝑡
𝑑𝑥=𝑣𝑑𝑡 ∫
𝑥0
𝑥
𝑑𝑥=¿∫
𝑡0
𝑡
𝑣𝑑𝑡¿ ∫
𝑥0
𝑥
𝑑𝑥=¿∫
𝑡0
𝑡
(𝑣0+𝑎𝑡)𝑑𝑡¿ 𝑥 − 𝑥0=𝑣0𝑡 +
1
2
𝑎𝑡
2
𝒙=𝒙𝟎+𝒗𝟎𝒕+
𝟏
𝟐
𝒂𝒕
𝟐
𝑎=
𝑑𝑣
𝑑𝑡
=
𝑑𝑣
𝑑 𝑥
𝑑 𝑥
𝑑𝑡
=𝑣
𝑑𝑣
𝑑𝑥
𝑣 𝑑𝑣=𝑎𝑑𝑥 ∫
𝑣0
𝑣
𝑣𝑑𝑣=¿∫
𝑥0
𝑥
𝑎𝑑𝑥¿
[𝑣
2
2 ]𝑣0
𝑣
=[𝑎𝑥]𝑥0
𝑥 𝑣2
− 𝑣0
2
2
=𝑎(𝑥− 𝑥0 )
𝒗𝟐
=𝒗𝟎
𝟐
+𝟐𝒂( 𝒙 −𝒙𝟎)

26. Free Fall
We use the term “free fall” to objects that are dropped down only guided by
acceleration due to gravity excluding air resistance.
We shall denote the magnitude of the acceleration due to gravity by the symbol g,
which is equal to 9.8 m/s2
near the Earth’s surface
we can make them simpler to use with the following minor changes:
1. The motion is along the vertical y-axis.
2. The free-fall acceleration is negative if the y-axis is chosen to be upward, and
hence we replace the acceleration a with -g.
3. It is positive if the y-axis is chosen to be downward, and hence we replace the
acceleration a with +g.

27. Equations of Motion fro Free Fall
y↑ y↓
At a given location on the Earth and in the absence of air resistance, all objects fall with the same constant
acceleration.
Acceleration due to gravity m/s2

28. 25.0
𝑚
𝑣0=20𝑚/ 𝑠
A ball is thrown vertically upwards with a velocity of 20 ms–1
from the top of a multistorey building. The
height of the point from where the ball is thrown is 25.0 m from the ground. (a) How high will the ball
rise? and (b) how long will it be before the ball hits the ground? Take ms–2
.
𝑔=10
𝑚
/
𝑠
2
Answer
At s, velocity of ball,and position of ball,
For upward motion,
At maximum height, velovity of ball,
If the ball rises to height y from the point of launch, then using the equation
𝑣2
=𝑣0
2
− 2𝑔( 𝑦− 𝑦0 ) 02
=202
− 2(10)(𝑦 − 𝑦0) 02
=4 00− 2 0 (𝑦 − 𝑦0)
( 𝑦− 𝑦0)=20 𝑚
(a)
(b)
𝑦= 𝑦0+𝑣0 𝑡 −
1
2
𝑔𝑡
2
0=25+20𝑡 −
1
2
(10 𝑡
2
) 20+25𝑡 −5𝑡2
=0 4+5𝑡 − 𝑡2
=0
𝑡2
−5𝑡 − 4=0 (𝑡 −5)(𝑡+1)=0 𝑡=5 𝑡=−1
and

29. Relative Velocity
Each plane is nearly at rest relative to the other, though both
are moving with very large velocities relative to Earth.
The surface of Earth, air outside the plane, and the plane itself are frames of reference. A frame of
reference is an extended object or collection of objects whose parts are at rest relative to each other. To
specify the velocity of an object requires that you specify the frame of reference that the velocity is
relative to.
Each observer, equipped in principle with a meter stick and a stopwatch, forms what we call a frame of
reference. Thus a frame of reference is a coordinate system plus a time scale.

30. Relative Velocity
𝑣𝐴=
𝑥𝐴 (𝑡) − 𝑥𝐴(0)
𝑡
𝑥𝐴(𝑡 )=𝑥𝐴 (0)+𝑣𝐴𝑡
𝑥𝐵 (𝑡)=𝑥𝐵 (0)+𝑣𝐵 𝑡
Similarly,
Displacement of B with respect to A,
𝑥𝐵𝐴 (𝑡)=𝑥𝐵 (𝑡)−𝑥𝐴(𝑡 )=𝑥𝐵 (0)+𝑣𝐵𝑡 −𝑥𝐴 (0)−𝑣𝐴 𝑡
𝑥𝐵𝐴 (𝑡)=[𝑥𝐵 (0)− 𝑥𝐴 (0) ]+[𝑣𝐵− 𝑣𝐴]𝑡
If both A and B start from the origin,
𝑥𝐵𝐴 (𝑡)=[𝑣𝐵 −𝑣𝐴]𝑡
𝑥 𝐵𝐴(𝑡 )
𝑡
=𝑣𝐵− 𝑣𝐴
𝑣𝐵 𝐴=𝑣𝐵 −𝑣𝐴
𝑣𝐴 𝐵=𝑣𝐴 −𝑣𝐵
𝑣𝐵𝐴=−𝑣𝐴𝐵

31. Relative Velocity
𝑥
𝑡
If
𝑥
𝑡
If
𝐴
𝐵
𝐴
𝐵
𝑥
𝑡
Ifandopposite signs
𝐴
𝐵

32. Thank You