Page # 2 KINEMATICS
KINEMATICS
In this lesson we shall study the geometry of motion i.e., kinematics.
Kinematics is used to relate displacement, velocity, acceleration
and time without reference to the cause of motion. We shall discuss
about kinematics of the particle here. Use of the word 'particles'
does not mean that our study will be restricted to small corpuscles;
rather, it indicates that in this lesson the motion of bodies possibly
as large as men, cars, rockets, or aeroplanes will be considered
without regard to their shape and size.
The entire lesson is divided into two sections. In the first
section we shall study about the motion in a straight line. In the
second section motion of particle in a plane specially projectile
motion and the concept of the relative motion of one particle with
respect to another will be discussed.
IIT-JEE Syllabus :
Kinematics in one and two dimension (cartesion coordinates only), projectiles;
Relative Motion.
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
, KINEMATICS Page # 3
1. REST AND MOTION :
* An object is said to be in motion wrt a frame of reference S1, when its location is changing with
time in same frame of reference S1.
* Rest and motion are relative terms.
* Absolute rest and absolute motion have no meaning.
Motion is broadly classified into 3 categories.
1. Rectilinear and translatory motion.
2. Circular and rotatory motion.
3. Oscillatory and vibratory motion.
1.1 Rectilinear or 1-D Motion
When a particle is moving along a straight line, then its motion is a rectilinear motion.
Parameters of rectilinear motion or translatory motion or plane motion :
(A) Time :
* It is a scalar quantity and its SI unit is second(s).
* At a particular instant of time, a physical object can be present at one location only.
* Time can never decrease.
y
(B) Position or location - It is defined with respect to A C
some reference point (origin) of given frame of reference.
r1 B
Consider a particle which moves from location r1 (at time t1)
r2
to location r2 (at time t2) as shown in the figure below,
following path ACB. x
(C) Distance :
The length of the actual path traversed by the particle is termed as its distance.
Distance = length of path ACB.
* Its SI unit is metre and it is a scalar quantity.
* It can never decrease with time.
(D) Displacement :
The change in position vector of the particle for a given time interval is known as its displacement.
→
AB = r = r2 − r1
* Displacement is a vector quantity and its SI unit is metre.
* It can decrease with time.
For a moving particle in a given interval of time
* Displacement can be +ve, –ve or 0, but distance would be always +ve.
* Distance ≥ Magnitude of displacement.
* Distance is always equal to displacement only and only if particle is moving along a straight line
without any change in direction.
(E) Average speed and average velocity :
Average speed and average velocity are always defined for a time interval.
Total dis tan ce travelled ∆s
Average speed(vav ) = =
Time int erval ∆t
Displacement ∆r r −r
Average velocity (vav ) = = = 2 1
Time int erval ∆t t2 − t1
* Average speed is a scalar quantity, while average velocity is a vector quantity. Both have the same
SI units, i.e., m/s.
For a moving particle in a given interval of time
* Average speed can be a many valued function but average velocity would be always a single-
valued function.
* Average velocity can be positive, negative or 0 but average speed would be always positive.
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
, Page # 4 KINEMATICS
(F) Instantaneous speed and instantaneous velocity
Instantaneous speed is also defined exactly like average speed i.e. it is equal to the ratio of total
distance and time interval, but with one qualification that time interval is extremely (infinitesimally)
small. The instantaneous speed is the speed at a particular instant of time and may have entirly
different value than that of average speed. Mathematically.
∆s ds S
v = lim = ...(4)
∆s →0 ∆t dt
When ∆s is the distance travelled in time ∆t. B
Distance
As ∆t tends to zero, the ratio defining speed becomes
finite and equals to the first derivative of the distance.
The speed at the moment 't' is is called the instantaneous D ∆S
speed at time 't'. θ
On the distance - time plot, the speed is equal to the slope A C
∆t
of the tangent to the curve at the time instant 't'. Let A t
and B point on the plot corresponds to the time t and t + O t t + ∆t time
∆t during the motion. As ∆ t approaches zero, the chord AB
becomes the tangent AC at A. The slope of the tangent Instantaneous speed is equal to the slope
of the tangent at given instant.
equal ds/dt, which is equal to the intantaneous speed at
't'.
DC ds
v = tanθ = =
AC dt
(G) Instantaneous velocity :
Instantaneous velocity is defined exactly like speed. It is
equal to the ratio of total displacement and time interval,
but with one qualification that time interval is extremely
(infinitesimally) small. Thus, instantaneous velocity can S
be termed as the average velocity at a particular instant
Position/displacement
of time when ∆ t tend to zero and may have entirely
B
different value that of average velocity : Mathematically.
B'
∆r dr D ∆S
v = lim = θ
∆t →0 ∆t dt
A ∆t C
As ∆ t tends to zero, the ratio defining velocity becomes t
O
finite and equals to the first derivative of the position t t + ∆t time
vector. The velocity at the moment 't' is called the Instantaneous velocity is equal to the slope
instantaneous velocity or simply velocity at time 't'. of the tangent at given instant.
The magnitude of average velocity |vavg| and average speed vavg may not be equal, but magnitude of
instantaneous velocity |v| is always equal to instantaneous speed v.
Ex.1 In 1.0 sec a particle goes from point A to point B moving in a semicircle of radius 1.0 m. The
magnitude of average velocity is
(A) 3.14 m/sec (B) 2.0 m/sec (C) 1.0 m/sec (D) zero
Total displacement d A
Sol. Average velocity = = 1m
Total time t o
D = AO + OB 1m
= 1 + 1 = 2m B
t = 1 sec (given)
2
⇒ mg of v of = 2m/sec
1
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
, KINEMATICS Page # 5
Ex.2 A particle moves along a semicircular path of radius R in time t with
constant speed. For the particle calculate
(i) distance travelled, R
A B
(ii) displacement,
(iii) average speed,
(iv) average velocity,
Sol. (i) Distance = length of path of particle = AB = πR
(ii) Displacement = minimum distance between initial and final point
= AB = 2R
total dis tan ce πR
(iii) Average speed, v = =
time t
2R
(iv) Average velocity =
t
Ex.3 A body travels the first half of the total distance with velocity v1 and the second half with
velocity v2. Calculate the average velocity :
Sol. Let total distance = 2x. Then
x x v1 + v 2 2x 2v1v 2
total time taken = v + v = x v v ∴ Average speed = = v +v
1 2 1 2 v1 + v 2 1 2
x
v1v 2
(G-1) When velocity is given as a function of t :
Ex.4 Velocity-time equation of a particle moving in a straight line is,
v = (10 + 2t + 3t2)
Find :
(a) displacement of particle from the origin of time t = 1 s, if it is given that displacement is 20 m at
time t = 0
(b) acceleration-time equation.
Sol. (a) The given equation can be written as,
ds
v= = (10 + 2t + 3t 2 )
dt
ds = (10 + 2t + 3t2) dt
s t
or ∫
20
∫
ds = (10 + 2t + 3t 2 )dt
0
or s – 20 = [10t + t2 + t3]01
or s = 20 + 12 = 32 m
(b) Acceleration-time equation can be obtained by differentiating the given equation w.r.t. time.
Thus,
dv d
a= = (10 + 2t + 3 t 2 ) or a = 2 + 6t
dt dt
SPECIMEN PROBLEM :
(A) WHEN EQUATION OF DISPLACEMENT IS GIVEN AND SPEED TO BE FIND OUT
Ex.5 If displacement is depend on time such that
x = 2t –2 then find out average speed upt to 4 sec.
Total distance
Sol. Average speed =
Total time
for Total distance
at t = 0 it is at x = – 2
at t = 1 it is at 0 m
at t = 4 it is at 6 m.
Total distance = |– 2| + 0 + 6 = 8 m
Average speed = 8/4 = 2m/sec
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
KINEMATICS
In this lesson we shall study the geometry of motion i.e., kinematics.
Kinematics is used to relate displacement, velocity, acceleration
and time without reference to the cause of motion. We shall discuss
about kinematics of the particle here. Use of the word 'particles'
does not mean that our study will be restricted to small corpuscles;
rather, it indicates that in this lesson the motion of bodies possibly
as large as men, cars, rockets, or aeroplanes will be considered
without regard to their shape and size.
The entire lesson is divided into two sections. In the first
section we shall study about the motion in a straight line. In the
second section motion of particle in a plane specially projectile
motion and the concept of the relative motion of one particle with
respect to another will be discussed.
IIT-JEE Syllabus :
Kinematics in one and two dimension (cartesion coordinates only), projectiles;
Relative Motion.
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
, KINEMATICS Page # 3
1. REST AND MOTION :
* An object is said to be in motion wrt a frame of reference S1, when its location is changing with
time in same frame of reference S1.
* Rest and motion are relative terms.
* Absolute rest and absolute motion have no meaning.
Motion is broadly classified into 3 categories.
1. Rectilinear and translatory motion.
2. Circular and rotatory motion.
3. Oscillatory and vibratory motion.
1.1 Rectilinear or 1-D Motion
When a particle is moving along a straight line, then its motion is a rectilinear motion.
Parameters of rectilinear motion or translatory motion or plane motion :
(A) Time :
* It is a scalar quantity and its SI unit is second(s).
* At a particular instant of time, a physical object can be present at one location only.
* Time can never decrease.
y
(B) Position or location - It is defined with respect to A C
some reference point (origin) of given frame of reference.
r1 B
Consider a particle which moves from location r1 (at time t1)
r2
to location r2 (at time t2) as shown in the figure below,
following path ACB. x
(C) Distance :
The length of the actual path traversed by the particle is termed as its distance.
Distance = length of path ACB.
* Its SI unit is metre and it is a scalar quantity.
* It can never decrease with time.
(D) Displacement :
The change in position vector of the particle for a given time interval is known as its displacement.
→
AB = r = r2 − r1
* Displacement is a vector quantity and its SI unit is metre.
* It can decrease with time.
For a moving particle in a given interval of time
* Displacement can be +ve, –ve or 0, but distance would be always +ve.
* Distance ≥ Magnitude of displacement.
* Distance is always equal to displacement only and only if particle is moving along a straight line
without any change in direction.
(E) Average speed and average velocity :
Average speed and average velocity are always defined for a time interval.
Total dis tan ce travelled ∆s
Average speed(vav ) = =
Time int erval ∆t
Displacement ∆r r −r
Average velocity (vav ) = = = 2 1
Time int erval ∆t t2 − t1
* Average speed is a scalar quantity, while average velocity is a vector quantity. Both have the same
SI units, i.e., m/s.
For a moving particle in a given interval of time
* Average speed can be a many valued function but average velocity would be always a single-
valued function.
* Average velocity can be positive, negative or 0 but average speed would be always positive.
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
, Page # 4 KINEMATICS
(F) Instantaneous speed and instantaneous velocity
Instantaneous speed is also defined exactly like average speed i.e. it is equal to the ratio of total
distance and time interval, but with one qualification that time interval is extremely (infinitesimally)
small. The instantaneous speed is the speed at a particular instant of time and may have entirly
different value than that of average speed. Mathematically.
∆s ds S
v = lim = ...(4)
∆s →0 ∆t dt
When ∆s is the distance travelled in time ∆t. B
Distance
As ∆t tends to zero, the ratio defining speed becomes
finite and equals to the first derivative of the distance.
The speed at the moment 't' is is called the instantaneous D ∆S
speed at time 't'. θ
On the distance - time plot, the speed is equal to the slope A C
∆t
of the tangent to the curve at the time instant 't'. Let A t
and B point on the plot corresponds to the time t and t + O t t + ∆t time
∆t during the motion. As ∆ t approaches zero, the chord AB
becomes the tangent AC at A. The slope of the tangent Instantaneous speed is equal to the slope
of the tangent at given instant.
equal ds/dt, which is equal to the intantaneous speed at
't'.
DC ds
v = tanθ = =
AC dt
(G) Instantaneous velocity :
Instantaneous velocity is defined exactly like speed. It is
equal to the ratio of total displacement and time interval,
but with one qualification that time interval is extremely
(infinitesimally) small. Thus, instantaneous velocity can S
be termed as the average velocity at a particular instant
Position/displacement
of time when ∆ t tend to zero and may have entirely
B
different value that of average velocity : Mathematically.
B'
∆r dr D ∆S
v = lim = θ
∆t →0 ∆t dt
A ∆t C
As ∆ t tends to zero, the ratio defining velocity becomes t
O
finite and equals to the first derivative of the position t t + ∆t time
vector. The velocity at the moment 't' is called the Instantaneous velocity is equal to the slope
instantaneous velocity or simply velocity at time 't'. of the tangent at given instant.
The magnitude of average velocity |vavg| and average speed vavg may not be equal, but magnitude of
instantaneous velocity |v| is always equal to instantaneous speed v.
Ex.1 In 1.0 sec a particle goes from point A to point B moving in a semicircle of radius 1.0 m. The
magnitude of average velocity is
(A) 3.14 m/sec (B) 2.0 m/sec (C) 1.0 m/sec (D) zero
Total displacement d A
Sol. Average velocity = = 1m
Total time t o
D = AO + OB 1m
= 1 + 1 = 2m B
t = 1 sec (given)
2
⇒ mg of v of = 2m/sec
1
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
, KINEMATICS Page # 5
Ex.2 A particle moves along a semicircular path of radius R in time t with
constant speed. For the particle calculate
(i) distance travelled, R
A B
(ii) displacement,
(iii) average speed,
(iv) average velocity,
Sol. (i) Distance = length of path of particle = AB = πR
(ii) Displacement = minimum distance between initial and final point
= AB = 2R
total dis tan ce πR
(iii) Average speed, v = =
time t
2R
(iv) Average velocity =
t
Ex.3 A body travels the first half of the total distance with velocity v1 and the second half with
velocity v2. Calculate the average velocity :
Sol. Let total distance = 2x. Then
x x v1 + v 2 2x 2v1v 2
total time taken = v + v = x v v ∴ Average speed = = v +v
1 2 1 2 v1 + v 2 1 2
x
v1v 2
(G-1) When velocity is given as a function of t :
Ex.4 Velocity-time equation of a particle moving in a straight line is,
v = (10 + 2t + 3t2)
Find :
(a) displacement of particle from the origin of time t = 1 s, if it is given that displacement is 20 m at
time t = 0
(b) acceleration-time equation.
Sol. (a) The given equation can be written as,
ds
v= = (10 + 2t + 3t 2 )
dt
ds = (10 + 2t + 3t2) dt
s t
or ∫
20
∫
ds = (10 + 2t + 3t 2 )dt
0
or s – 20 = [10t + t2 + t3]01
or s = 20 + 12 = 32 m
(b) Acceleration-time equation can be obtained by differentiating the given equation w.r.t. time.
Thus,
dv d
a= = (10 + 2t + 3 t 2 ) or a = 2 + 6t
dt dt
SPECIMEN PROBLEM :
(A) WHEN EQUATION OF DISPLACEMENT IS GIVEN AND SPEED TO BE FIND OUT
Ex.5 If displacement is depend on time such that
x = 2t –2 then find out average speed upt to 4 sec.
Total distance
Sol. Average speed =
Total time
for Total distance
at t = 0 it is at x = – 2
at t = 1 it is at 0 m
at t = 4 it is at 6 m.
Total distance = |– 2| + 0 + 6 = 8 m
Average speed = 8/4 = 2m/sec
394,50 - Rajeev Gandhi Nagar Kota, Ph. No. : 93141-87482, 0744-2209671
IVRS No : 0744-2439051, 52, 53, www. motioniitjee.com ,
www.rancho.in
