Motion in a straight line

© 2013 Pearson Education, Inc.
.
Slide 2-2

 Kinematics is the study of motion
without going into its causes.
 This chapter deals with motion
along a straight line, i.e.
rectilinear motion.
 The motion is the change in position
of an object with respect to time.
Slide 2-3

 The distance an object travels is a scalar quantity,
independent of direction.
 The displacement of an object is a vector quantity,
equal to the final position minus the initial position.
 An object’s speed v is scalar quantity, independent
of direction.
 Speed is how fast an object is going; it is always
positive.
 Velocity is a vector quantity that includes direction.
 In one dimension the direction of velocity is specified
by the + or − sign.
Slide 2-28

 Distance = length of the actual
path taken to go from source to
destination
 Displacement = length of the
straight line joining the source
to the destination or in other
words the length of the
shortest path

Example: If a boy
walks from B to D
[arc] in a circular
path, the distance
will be the
semicircle of the
circle, while the
displacement will be
the diameter BD.
B
A
C
D

 Rohit and Seema both start from their house. Rohit
walks 2 km to the east while Seema walk 1 km to the
west and then turns back and walks 1 km.
Seema is back home and her displacement is 0 m.
This is because direction of motion is different in
both cases.
You require both distance and direction to
determine displacement.
Distance travelled by them is the same (2 km)

 Uniform motion is when
equal displacements occur
during any successive
equal-time intervals.
 Uniform motion
is always along
a straight line.
 Eg; While driving a car at a
perfectly steady 60 kmph,
this means there is a
change in the position by 60
km for every time interval
of 1 hour.
Slide 2-20
Riding steadily over level ground is a
good example of uniform motion.

 An object’s motion is
uniform if and only if its
position-versus-time
graph is a straight line.
 The average velocity is
the slope of the position-
versus-time graph.
 The SI units of velocity
are m/s.
Slide 2-21

25
20
15
10
5
0
10 20 30 40 500
Time (s)
Distance – Time graph

1.25
1.0
0.75
0.5
0.25
0
10 20 30 40 500
Time (s)
Speed – Time graph

1.25
1.0
0.75
0.5
0.25
0
10 20 30 40 500
Time (s)
Velocity – Time graph
Non-uniform Motion
Acceleration = 0.125 m/s2

 An object that is speeding up or slowing down is not in
uniform motion.
 In this case, the position-versus-time graph is not a
straight line.
 We can determine the average speed Vav between any two
times separated by time interval ∆t by finding the slope
of the straight-line connection between the two points.
 The instantaneous velocity is the object’s velocity
at a single instant of time t.
 The average velocity Vav = ∆s/∆t becomes a better and
better approximation to the instantaneous velocity as ∆t
gets smaller and smaller.
Slide 2-31

© 2013 Pearson Education, Inc. Slide 2-32
Motion diagrams and position graphs of an accelerating
rocket.

 As ∆t continues to get smaller, the average velocity
vavg = ∆s/∆t reaches a constant or limiting value.
 The instantaneous velocity at time t is the average
velocity during a time interval ∆t centered on t, as ∆t
approaches zero.
 In calculus, this is called the derivative of s with respect to t.
 Graphically, ∆s/∆t is the slope of a straight line.
 In the limit ∆t → 0, the straight line is tangent to the curve.
 The instantaneous velocity at time t is the slope of the line
that is tangent to the position-versus-time graph at time t.
Slide 2-33

 ds/dt is called the derivative of s with respect to t.
 ds/dt is the slope of the line that is tangent to the
position-versus-time graph.
 Consider a function u that depends on time as
u = ctn
, where c and n are constants:
 The derivative of a constant is zero:
 The derivative of a sum is the sum of the derivatives.
If u and w are two separate functions of time, then:
Slide 2-46

 Taking the derivative of a function is equivalent to
finding the slope of a graph of the function.
 Similarly, evaluating an integral is equivalent to
finding the area under a graph of the function.
 Consider a function u that depends on time as u = ctn
,
where c and n are constants:
 The vertical bar in the third step means the integral
evaluated at tf minus the integral evaluated at ti.
 The integral of a sum is the sum of the integrals. If u
and w are two separate functions of time, then:
Slide 2-60

 The SI units of acceleration are (m/s)/s, or m/s2
.
 It is the rate of change of velocity and measures how
quickly or slowly an object’s velocity changes.
 The average acceleration during a time interval ∆t is:
 Graphically, aavg is the slope of a straight-line velocity-
versus-time graph.
 If acceleration is constant, the acceleration as is the
same as aavg.
 Acceleration, like velocity, is a vector quantity and has
both magnitude and direction. Slide 2-64

1.25
1.0
0.75
0.5
0.25
0
10 20 30 40 500
Time (s)
Uniform Acceleration
Acceleration = 0.125 m/s2

1.25
1.0
0.75
0.5
0.25
0
10 20 30 40 500
Time (s)
Non-uniform Acceleration

v
u
0
t
0
Time (s)
Initial velocity = u
Final velocity = v
Time = t
Acceleration = a
Displacement = s

v
u
0
t0
Time (s)
Initial velocity = u
Final velocity = v
Time = t
Acceleration = a
Displacement = s

Derivation of equation 1; atvv += 0
dt
dv
a = a is constant → adtdv =
Integrate both sides with respect to time from 0
to t
atvvatv
dtaadtvdt
v
v
v t t
+=⇒=
==∫ ∫ ∫
0
0 0 0
0
][

[ ] [ ]
t
0 0
0 0
2
0
Derivation of eqs.2
( )
Integrate both sides with respect to time from 0 to t
dx
2
o o
o
t t
t t
o
o o
o
dx
v dx vdt v v at dx v at dt
dt
dx v dt atdt
v
at
x x
dt a tdt v
t
t
v
x
= → = = + → = + →
= +
= + → =
= + +
∫ ∫ ∫
2
0
2 2
2
( 0) 0
2 2
t
o o o
t
a
t t
x x v t a v t a
 
+ → 
 
 
− = − + − = + → 
 
2
2
o o
at
x x v t= + +
x x

a = dv/dt=dv/dx × dx/dt
a =vdv/dx
vdv= adx
x x
∫u vdv = ∫x adx
v 2
- u2
= a ( x- x0 ) =as
─────
2
v2
– u2
= 2as

 The motion of an object moving
under the influence of gravity
only, and no other forces, is
called free fall.
 Two objects dropped from the
same height will, if air
resistance can be neglected, hit
the ground at the same time and
with the same speed.
 Consequently, any two objects
in free fall, regardless of their
mass, have the same
acceleration:
Slide 2-94
In the absence of air resistance, any two
objects fall at the same rate and hit the
ground at the same time. The apple and
feather seen here are falling in a vacuum.

 Figure (a) shows the motion
diagram of an object that was
released from rest and falls freely.
 Figure (b) shows the object’s
velocity graph.
 The velocity graph is a straight
line with a slope:
 Where g is a positive number
which is equal to 9.80 m/s2
on
the surface of the earth.
 Other planets have different values of g.
Slide 2-95

 When two bodies are moving in the same
direction parallel to each other;
vab = va – vb
 When two bodies are moving in opposite
directions;
vab = va + vb
 When two bodies make an angle with each
other
vab = /va2
+ vb2
+ 2vabcos

The bus moved away from the tree
The person is comparing the position of the bus with respect
to the position of the tree
Reference (or origin) is position of the tree
0 5 10

The tree moved away from the bus.
The person is comparing the position of the tree with respect
to the position of the bus.
Reference (or origin) is position of the bus.
0510

Both the observations are correct. The difference is what is
taken as the origin.
Motion is always relative. When one says that a object is
moving, he/she is comparing the position of that object with
another object.
Motion is therefore change in position of an object with
respect to another object over time.
Kinematics studies motion without delving into what caused
the motion.

Motion in a straight line

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