Kinematics 1-D | Physics Optional Notes for UPSC PDF Download

 Page 1


KINEMATICS 1-D 
KINEMATICS 
Kinematics deals with study of motion of a particle without any regard to cause of 
motion. 
1. FRAME OF REFERENCE 
 
We can only see something moving if it's changing its position compared to something 
else. So, for us to notice motion, there has to be two things: one thing that's moving, and 
another thing it's moving in relation to. And of course, there's someone watching this 
happen – that's the observer. 
In this way, motion of the moving body is expressed in terms of its position coordinates 
changing with time. 
2. MOTION & REST 
If a body changes its position with time, it is said to be moving else it is at rest. 
Motion/rest is always relative to the observer. 
Motion/rest are a combined property of the object under study and the observer. There 
is no meaning of rest or motion without the observer or frame of reference. 
? If only one coordinate change with time, motion is one dimensional motion ( 1 - 
D) or straight line motion. 
If only two coordinates change with time, motion is two dimensional (2-D) or motion in 
a plane. If all three coordinates changes with time, motion is three dimensional (3 - D) or 
motion in space. 
? The reference frame is chosen according to problems. 
? If frame is not mentioned, then ground is taken as reference frame. 
3. DISTANCE & DISPLACEMENT 
Distance 
Total length of path covered by the particle, in definite time interval. 
Let a body moves from ?? to ?? via ?? . The length of path ?????? is called the distance 
travelled by the body. 
Page 2


KINEMATICS 1-D 
KINEMATICS 
Kinematics deals with study of motion of a particle without any regard to cause of 
motion. 
1. FRAME OF REFERENCE 
 
We can only see something moving if it's changing its position compared to something 
else. So, for us to notice motion, there has to be two things: one thing that's moving, and 
another thing it's moving in relation to. And of course, there's someone watching this 
happen – that's the observer. 
In this way, motion of the moving body is expressed in terms of its position coordinates 
changing with time. 
2. MOTION & REST 
If a body changes its position with time, it is said to be moving else it is at rest. 
Motion/rest is always relative to the observer. 
Motion/rest are a combined property of the object under study and the observer. There 
is no meaning of rest or motion without the observer or frame of reference. 
? If only one coordinate change with time, motion is one dimensional motion ( 1 - 
D) or straight line motion. 
If only two coordinates change with time, motion is two dimensional (2-D) or motion in 
a plane. If all three coordinates changes with time, motion is three dimensional (3 - D) or 
motion in space. 
? The reference frame is chosen according to problems. 
? If frame is not mentioned, then ground is taken as reference frame. 
3. DISTANCE & DISPLACEMENT 
Distance 
Total length of path covered by the particle, in definite time interval. 
Let a body moves from ?? to ?? via ?? . The length of path ?????? is called the distance 
travelled by the body. 
 
But overall, the body is displaced from ?? to ?? . A vector from ?? to ?? , i.e. ????
????? 
 is its 
displacement vector or displacement that is the minimum distance and directed from the 
initial position to the final position. 
Displacement in terms of position vector 
Let a body be displaced from A(x
1
,y
1
,z
1
) to B(x
2
,y
2
,z
2
) then its displacement is given by 
vector AB
????? 
. 
 From ?OAB r 
A
+ ?r = r 
B
  or  ?r = r 
B
- r 
A
? r 
B
= x
2
i ˆ + y
2
??ˆ + ?? 2
?? ˆ
   and  ?? 
A
= x
1
i ˆ + y
1
??ˆ + ?? 1
?? ˆ
? ?r = (x
2
- x
1
)i ˆ + (y
2
- y
1
)j ˆ + (?? 2
- ?? 1
)?? ˆ
  or  ?r = ??? i ˆ + ??? ??ˆ + ??? k
ˆ
 
 
 
(eg.) 
Distance 
Displacement 
A ? B
?? R
2
 A ? C ?? R
 
v2R in N - E 
2R in N 
 
Direction must be satisfied in displacement. 
Average speed: 
i.e. average speed is rate of travelling distance over a time interval. 
Page 3


KINEMATICS 1-D 
KINEMATICS 
Kinematics deals with study of motion of a particle without any regard to cause of 
motion. 
1. FRAME OF REFERENCE 
 
We can only see something moving if it's changing its position compared to something 
else. So, for us to notice motion, there has to be two things: one thing that's moving, and 
another thing it's moving in relation to. And of course, there's someone watching this 
happen – that's the observer. 
In this way, motion of the moving body is expressed in terms of its position coordinates 
changing with time. 
2. MOTION & REST 
If a body changes its position with time, it is said to be moving else it is at rest. 
Motion/rest is always relative to the observer. 
Motion/rest are a combined property of the object under study and the observer. There 
is no meaning of rest or motion without the observer or frame of reference. 
? If only one coordinate change with time, motion is one dimensional motion ( 1 - 
D) or straight line motion. 
If only two coordinates change with time, motion is two dimensional (2-D) or motion in 
a plane. If all three coordinates changes with time, motion is three dimensional (3 - D) or 
motion in space. 
? The reference frame is chosen according to problems. 
? If frame is not mentioned, then ground is taken as reference frame. 
3. DISTANCE & DISPLACEMENT 
Distance 
Total length of path covered by the particle, in definite time interval. 
Let a body moves from ?? to ?? via ?? . The length of path ?????? is called the distance 
travelled by the body. 
 
But overall, the body is displaced from ?? to ?? . A vector from ?? to ?? , i.e. ????
????? 
 is its 
displacement vector or displacement that is the minimum distance and directed from the 
initial position to the final position. 
Displacement in terms of position vector 
Let a body be displaced from A(x
1
,y
1
,z
1
) to B(x
2
,y
2
,z
2
) then its displacement is given by 
vector AB
????? 
. 
 From ?OAB r 
A
+ ?r = r 
B
  or  ?r = r 
B
- r 
A
? r 
B
= x
2
i ˆ + y
2
??ˆ + ?? 2
?? ˆ
   and  ?? 
A
= x
1
i ˆ + y
1
??ˆ + ?? 1
?? ˆ
? ?r = (x
2
- x
1
)i ˆ + (y
2
- y
1
)j ˆ + (?? 2
- ?? 1
)?? ˆ
  or  ?r = ??? i ˆ + ??? ??ˆ + ??? k
ˆ
 
 
 
(eg.) 
Distance 
Displacement 
A ? B
?? R
2
 A ? C ?? R
 
v2R in N - E 
2R in N 
 
Direction must be satisfied in displacement. 
Average speed: 
i.e. average speed is rate of travelling distance over a time interval. 
Average speed =
 Distance travelled 
 Time taken 
 
Average velocity: 
Average velocity =
 Displacement 
 Time taken 
 
i.e.,  ?v ? >=
?r ? 
?t
=
r ? 
f
-r ? 
i
t
f
-t
r
 
(e.g.) 
 Speed < ?? >
?? ? ?? ????
2
= ?? m/s
v2R
1
= 2v2 m/s in N - E
A ? C
?? R
2
= ?? m/s
2R
2
= 2 m/s in N
A ? A
2?? R
4
= ?? m/s 0
 
 
?? VERAGE AND INSTANTANEOUS RATE OF CHANGE 
Every vector quantity have two types of rate of change < Average 
Average: Average rate of change is defined for an interval. Average rate of change of y 
with respect to ?? is written as 
??? ??? 
Instantaneous: The instantaneous rate of change is defined for a given instant. 
Instantaneous rate of change of ?? with respect to (w.r.t.) ?? is written as 
????
????
 which is also 
known as differentiation of ?? with respect to ?? . 
(e.g.) Average Instantaneous 
< I >=
??? ?t
I =
d?? dt
< F
? 
>=
?P
? ? 
?t
F
? 
=
dP
? ? 
dt
< v ? >=
?r 
?t
v ? =
dr 
dt
 
Derivatives of Commonly Used Functions: 
Page 4


KINEMATICS 1-D 
KINEMATICS 
Kinematics deals with study of motion of a particle without any regard to cause of 
motion. 
1. FRAME OF REFERENCE 
 
We can only see something moving if it's changing its position compared to something 
else. So, for us to notice motion, there has to be two things: one thing that's moving, and 
another thing it's moving in relation to. And of course, there's someone watching this 
happen – that's the observer. 
In this way, motion of the moving body is expressed in terms of its position coordinates 
changing with time. 
2. MOTION & REST 
If a body changes its position with time, it is said to be moving else it is at rest. 
Motion/rest is always relative to the observer. 
Motion/rest are a combined property of the object under study and the observer. There 
is no meaning of rest or motion without the observer or frame of reference. 
? If only one coordinate change with time, motion is one dimensional motion ( 1 - 
D) or straight line motion. 
If only two coordinates change with time, motion is two dimensional (2-D) or motion in 
a plane. If all three coordinates changes with time, motion is three dimensional (3 - D) or 
motion in space. 
? The reference frame is chosen according to problems. 
? If frame is not mentioned, then ground is taken as reference frame. 
3. DISTANCE & DISPLACEMENT 
Distance 
Total length of path covered by the particle, in definite time interval. 
Let a body moves from ?? to ?? via ?? . The length of path ?????? is called the distance 
travelled by the body. 
 
But overall, the body is displaced from ?? to ?? . A vector from ?? to ?? , i.e. ????
????? 
 is its 
displacement vector or displacement that is the minimum distance and directed from the 
initial position to the final position. 
Displacement in terms of position vector 
Let a body be displaced from A(x
1
,y
1
,z
1
) to B(x
2
,y
2
,z
2
) then its displacement is given by 
vector AB
????? 
. 
 From ?OAB r 
A
+ ?r = r 
B
  or  ?r = r 
B
- r 
A
? r 
B
= x
2
i ˆ + y
2
??ˆ + ?? 2
?? ˆ
   and  ?? 
A
= x
1
i ˆ + y
1
??ˆ + ?? 1
?? ˆ
? ?r = (x
2
- x
1
)i ˆ + (y
2
- y
1
)j ˆ + (?? 2
- ?? 1
)?? ˆ
  or  ?r = ??? i ˆ + ??? ??ˆ + ??? k
ˆ
 
 
 
(eg.) 
Distance 
Displacement 
A ? B
?? R
2
 A ? C ?? R
 
v2R in N - E 
2R in N 
 
Direction must be satisfied in displacement. 
Average speed: 
i.e. average speed is rate of travelling distance over a time interval. 
Average speed =
 Distance travelled 
 Time taken 
 
Average velocity: 
Average velocity =
 Displacement 
 Time taken 
 
i.e.,  ?v ? >=
?r ? 
?t
=
r ? 
f
-r ? 
i
t
f
-t
r
 
(e.g.) 
 Speed < ?? >
?? ? ?? ????
2
= ?? m/s
v2R
1
= 2v2 m/s in N - E
A ? C
?? R
2
= ?? m/s
2R
2
= 2 m/s in N
A ? A
2?? R
4
= ?? m/s 0
 
 
?? VERAGE AND INSTANTANEOUS RATE OF CHANGE 
Every vector quantity have two types of rate of change < Average 
Average: Average rate of change is defined for an interval. Average rate of change of y 
with respect to ?? is written as 
??? ??? 
Instantaneous: The instantaneous rate of change is defined for a given instant. 
Instantaneous rate of change of ?? with respect to (w.r.t.) ?? is written as 
????
????
 which is also 
known as differentiation of ?? with respect to ?? . 
(e.g.) Average Instantaneous 
< I >=
??? ?t
I =
d?? dt
< F
? 
>=
?P
? ? 
?t
F
? 
=
dP
? ? 
dt
< v ? >=
?r 
?t
v ? =
dr 
dt
 
Derivatives of Commonly Used Functions: 
? y = constant 
?
dy
dx
= 0 
? ?? = cos ?? 
?
????
????
= -sin ?? 
? y = x
n
 
?
dy
dx
= nx
n-1
 
? y = tan x 
?
????
????
= sec
2
 ?? 
? y = e
x
 
?
????
????
= ?? ?? 
? y = cot x 
?
dy
dx
= -cosec
2
 x 
? ?? = ln ?? 
?
dy
dx
=
1
x
 
? y = cosec x 
?
dy
dx
= -cosec xcot x 
? ?? = sin ?? 
?
????
????
= cos ?? 
? ?? = sec ?? 
?
????
????
= sec ?? tan ?? 
Method of Differentiation: 
If ?? = ?? (?? ) , let us denote 
????
????
= ?? '
(?? ) 
? Sum or Subtraction of two functions ?? = ?? (?? )± ?? (?? )?
????
????
= ?? '
(?? )± ?? '
(?? ) 
? Product of two functions 
?? = ?? (?? )· ?? (?? )?
????
????
= ?? (?? )?? '
(?? )+ ?? (?? )?? '
(?? ) 
? Division of two functions. 
?? =
?? (?? )
?? (?? )
?
????
????
=
?? (?? )?? '
(?? )- ?? (?? )?? '
(?? )
{?? (?? )}
2
 
Page 5


KINEMATICS 1-D 
KINEMATICS 
Kinematics deals with study of motion of a particle without any regard to cause of 
motion. 
1. FRAME OF REFERENCE 
 
We can only see something moving if it's changing its position compared to something 
else. So, for us to notice motion, there has to be two things: one thing that's moving, and 
another thing it's moving in relation to. And of course, there's someone watching this 
happen – that's the observer. 
In this way, motion of the moving body is expressed in terms of its position coordinates 
changing with time. 
2. MOTION & REST 
If a body changes its position with time, it is said to be moving else it is at rest. 
Motion/rest is always relative to the observer. 
Motion/rest are a combined property of the object under study and the observer. There 
is no meaning of rest or motion without the observer or frame of reference. 
? If only one coordinate change with time, motion is one dimensional motion ( 1 - 
D) or straight line motion. 
If only two coordinates change with time, motion is two dimensional (2-D) or motion in 
a plane. If all three coordinates changes with time, motion is three dimensional (3 - D) or 
motion in space. 
? The reference frame is chosen according to problems. 
? If frame is not mentioned, then ground is taken as reference frame. 
3. DISTANCE & DISPLACEMENT 
Distance 
Total length of path covered by the particle, in definite time interval. 
Let a body moves from ?? to ?? via ?? . The length of path ?????? is called the distance 
travelled by the body. 
 
But overall, the body is displaced from ?? to ?? . A vector from ?? to ?? , i.e. ????
????? 
 is its 
displacement vector or displacement that is the minimum distance and directed from the 
initial position to the final position. 
Displacement in terms of position vector 
Let a body be displaced from A(x
1
,y
1
,z
1
) to B(x
2
,y
2
,z
2
) then its displacement is given by 
vector AB
????? 
. 
 From ?OAB r 
A
+ ?r = r 
B
  or  ?r = r 
B
- r 
A
? r 
B
= x
2
i ˆ + y
2
??ˆ + ?? 2
?? ˆ
   and  ?? 
A
= x
1
i ˆ + y
1
??ˆ + ?? 1
?? ˆ
? ?r = (x
2
- x
1
)i ˆ + (y
2
- y
1
)j ˆ + (?? 2
- ?? 1
)?? ˆ
  or  ?r = ??? i ˆ + ??? ??ˆ + ??? k
ˆ
 
 
 
(eg.) 
Distance 
Displacement 
A ? B
?? R
2
 A ? C ?? R
 
v2R in N - E 
2R in N 
 
Direction must be satisfied in displacement. 
Average speed: 
i.e. average speed is rate of travelling distance over a time interval. 
Average speed =
 Distance travelled 
 Time taken 
 
Average velocity: 
Average velocity =
 Displacement 
 Time taken 
 
i.e.,  ?v ? >=
?r ? 
?t
=
r ? 
f
-r ? 
i
t
f
-t
r
 
(e.g.) 
 Speed < ?? >
?? ? ?? ????
2
= ?? m/s
v2R
1
= 2v2 m/s in N - E
A ? C
?? R
2
= ?? m/s
2R
2
= 2 m/s in N
A ? A
2?? R
4
= ?? m/s 0
 
 
?? VERAGE AND INSTANTANEOUS RATE OF CHANGE 
Every vector quantity have two types of rate of change < Average 
Average: Average rate of change is defined for an interval. Average rate of change of y 
with respect to ?? is written as 
??? ??? 
Instantaneous: The instantaneous rate of change is defined for a given instant. 
Instantaneous rate of change of ?? with respect to (w.r.t.) ?? is written as 
????
????
 which is also 
known as differentiation of ?? with respect to ?? . 
(e.g.) Average Instantaneous 
< I >=
??? ?t
I =
d?? dt
< F
? 
>=
?P
? ? 
?t
F
? 
=
dP
? ? 
dt
< v ? >=
?r 
?t
v ? =
dr 
dt
 
Derivatives of Commonly Used Functions: 
? y = constant 
?
dy
dx
= 0 
? ?? = cos ?? 
?
????
????
= -sin ?? 
? y = x
n
 
?
dy
dx
= nx
n-1
 
? y = tan x 
?
????
????
= sec
2
 ?? 
? y = e
x
 
?
????
????
= ?? ?? 
? y = cot x 
?
dy
dx
= -cosec
2
 x 
? ?? = ln ?? 
?
dy
dx
=
1
x
 
? y = cosec x 
?
dy
dx
= -cosec xcot x 
? ?? = sin ?? 
?
????
????
= cos ?? 
? ?? = sec ?? 
?
????
????
= sec ?? tan ?? 
Method of Differentiation: 
If ?? = ?? (?? ) , let us denote 
????
????
= ?? '
(?? ) 
? Sum or Subtraction of two functions ?? = ?? (?? )± ?? (?? )?
????
????
= ?? '
(?? )± ?? '
(?? ) 
? Product of two functions 
?? = ?? (?? )· ?? (?? )?
????
????
= ?? (?? )?? '
(?? )+ ?? (?? )?? '
(?? ) 
? Division of two functions. 
?? =
?? (?? )
?? (?? )
?
????
????
=
?? (?? )?? '
(?? )- ?? (?? )?? '
(?? )
{?? (?? )}
2
 
? Chain Rule 
?? = ?? {?? (?? )} ?
????
????
= ?? '
(?? )?? '
{?? (?? )} 
Example. Find 
????
????
, when (i) ?? = v?? (ii) ?? = ?? 5
+ ?? 4
+ 7  (iii) ?? = ?? 2
+ 4?? -1/2
- 3?? -2
 
Solution. (i) 
dy
dx
=
d
dx
(vx)=
d
dx
(x
1/2
)=
1
2
x
1/2-1
=
1
2
x
-1/2
=
1
2vx
 
(ii) 
????
????
=
?? ????
(?? 5
+ ?? 4
+ 7)=
?? ????
(?? 5
)+
?? ????
(?? 4
)+
?? ????
(7)= 5?? 4
+ 4?? 3
+ 0 = 5?? 4
+ 4?? 3
 
(iii) 
????
????
=
?? ????
(?? 2
+ 4?? -1/2
- 3?? -2
)=
?? ????
(?? 2
)+
?? ????
(4?? -1/2
)-
?? ????
(3?? -2
) 
=
d
dx
(x
2
)+ 4
d
dx
(x
-1/2
)- 3
d
dx
(x
-2
)= 2x+ 4(-
1
2
)x
-3/2
- 3(-2)x
-3
= 2x- 2x
-3/2
+ 6x
-3
 
Example. ?? =
?? 2
sin ?? 
Solution: 
????
????
=
sin (
?? ?? 2
????
)-?? 2
(
?? sin ?? ????
)
(sin ?? )
2
=
(sin ?? )2?? -?? 2
cos ?? (sin ?? )
2
 
Velocity: It is rate of change of position vector with time. 
 
Speed: Rate of travelling distance with time. 
 
KEY POINT 
1. |?? | = |??? |  (i.e. distance = displacement) 
2. 
s
?t
= |
?r ? 
?t
| 
Average speed >| average ?? |