Hone your preparation for IIT JEE Kinematics with askIITians best tips. Get top advice from ex-IITians for IIT JEE kinematics preparation

1. Kinematics for IIT JEE
Kinematics is an extremely important and simplest part of mechanics unit of physics for JEE
Advanced, JEE Main and other engineering exams. It is a
prerequisite for all other chapters of mechanics. This branch of
mechanics deals with description of motion of bodies and motion of
particle in one and two dimensions. Beginners are advised to refer
the study material on Kinematics.
Important topics in kinematics for IIT JEE:
It is an extremely vast topic and can be further divided into important parts like:
Definition of position vector, velocity and acceleration
Motion in 1-dimension (Rectilinear)
 Uniform acceleration
 Non – uniform acceleration (Calculus based)
 Graphs and application
Motion in 2-dimension (Curvilinear or Plane motion)
 Projectile
 Circular
 General Plane motion
It is important to master each and every topic of kinematics in order to gain excellence in the
mechanics portion of IIT JEE.
Motion
Change in position with respect to time is defined as motion.
Reference frame
It is the set of three coordinate axis attached to a particular observer who is observing the motion
of a particle or a body. If our reference frame is fixed in space, then it is known as Inertial frame
of reference and if reference frame is accelerating, then it is known as non-inertial frame of
reference. All the fundament laws of mechanics hold good in inertial frame only.

2. Position vector of a particle
With respect to a reference frame, the position vector of a particle is the vector whose tailpoints
toward the observer and head is at the object.
Let the observer be at O (rest) and particle is at P. Then, OP is the position vector of the particle.
Displacement vector
The change in a position vector is defined as the displacement vector. Let be the initial position
vector of the particle and be the final position vector of the particle. Then, displacement vector
is:
1-dimension & 2-dimension motion
When a particle is moving in such a way that direction of position vector does not change, then we
say it is 1-dimension, rectilinear or straight line motion and if the position vector is changing in such
a way that the displacement vector also changes its direction, then it is termed as 2-dimension,
curvilinear or motion in plane.
Velocity
Instantaneous Velocity
Let 푟⃗ be the position vector of a particle at any instant t. Then instantaneous velocity 푣⃗ is defined
as
푣⃗ =
푑푟⃗
푑푡
Note: Instantaneous speed is magnitude of instantaneous velocity.
Average Velocity
Average Speed
Average Speed = Distance/Time
⃗푣⃗⃗푎⃗⃗⃗푣⃗ =
Δ푟⃗
Δ푡
Note: Average velocity is in the direction of displacement vector.

3. Acceleration
Instantaneous Acceleration
It is defined as rate of change of velocity vector 푣⃗.
Average Acceleration
Motion in 1-dimension
Points to remember:
푎⃗ =
푑푣⃗
푑푡
⃗푎⃗⃗⃗푎⃗⃗푣⃗ =
Δ푣⃗
Δ푡
Velocity (푣⃗) is always along a fixed line.
Acceleration (푎⃗) is either parallel (accelerated motion) or anti-parallel (retarded motion) to
velocity (푣⃗).
Important formulas
Uniform acceleration
Velocity of particle after time t with initial velocity 푢⃗⃗ and uniform acceleration 푎⃗ is
푣⃗ = 푢⃗⃗ + 푎⃗푡
Displacement covered by particle in time t with initial velocity 푢⃗⃗ and uniform acceleration 푎⃗ is
푆⃗ = 푢⃗⃗푡 +
1
2
푎⃗푡2
Velocity of particle after covering displacement 푠⃗ with initial velocity 푢⃗⃗ and uniform acceleration 푎⃗
is
푣⃗2 − 푢⃗⃗2 = 2푎⃗푠⃗
Displacement covered in a 푛푡ℎ second with initial velocity 푢⃗⃗ and uniform acceleration 푎⃗ is
푆⃗푛 = 푢⃗⃗ +
1
2
푎⃗(2푛 − 1)

4. Non – uniform acceleration
Calculus Approach
Acceleration as a function of time
풂 = 풇(풕)
풅풗
풅풕
= 풇(풕)
∫ 풅풗 = ∫ 풇(풕)풅풕
From here, we will get velocity – time equation
풗 = 풇(풕)
풅풙
풅풕
= 풇(풕)
∫ 풅풙 = ∫ 풇(풕)풅풕
We will get the position – time equation.
Acceleration as a function of velocity (v⃗⃗)
풂 = 풇(풗)
풅풗
풅풕
= 풇(풗)
∫
풅풗
풇(풗)
= ∫ 풅풕
We will get velocity – time equation from here.
Similarly, we will get position – time equation by putting
풗 =
풅풙
풅풕
Acceleration as a function of position
풂 = 풇(풙)
풗
풅풗
풅풙
= 풇(풙)
∫ 풗풅풗 = ∫ 풇(풙)풅풙

5. We will get velocity-position equation from here
And by putting
풗 =
풅풙
풅풕
We will get position-time equation.
Position as a function of time
풙 = 풇(풕)
풗 =
풅풙
풅풕
풂 =
풅풗
풅풕
Velocity as a function of position
풗 = 풇(풙)
풅풙
풅풕
= 풇(풙)
∫
풅풙
풇(풙)
= ∫ 풅풕
We will get position-time equation from here and then repeat the working of previous case.
Velocity as a function of time
풗 = 풇(풕)
풅풙
풅풕
= 풇(풕)
∫ 풅풙 = ∫ 풇(풕)풅풕
We will get position-time equation from here
Acceleration-time equation
풂 =
풅풗
풅풕

6. Motion in 2-dimension
Circular Motion
Angular Displacement (Δ휃): Change in angular position of a particle.
Δ휽 = 휽풇 − 휽풊
Angular Velocity (휔):
흎 =
풅휽
풅풕
흎풂풗풆. =
휽풇 − 휽풊
풕
Angular Acceleration (α):
휶 =
풅흎
풅풕
휶풂풗풆. =
흎풇 − 흎풊
풕
Relation b/w linear and angular variables
풍 = 풓휽
풗⃗⃗⃗ = 흎⃗⃗⃗⃗ × 풓⃗⃗
Case I: Uniform circular motion: Motion with constant speed
Velocity vector is always along the tangent
Acceleration is directed toward the center and is known as centripetal acceleration (푎⃗⃗⃗⃗푐⃗)
Angular velocity is constant and angular acceleration is zero
Let the position vector be 푟⃗ and 휃 be angular position w.r.t. to some reference line, then
Velocity, v =
dr
dt
Speed = Magnitude of v = r
dθ
dt
Acceleration, a =
dv
dt
Magnitude of a = v
dθ
dt

7. Case II: Non-uniform circular motion: Motion with varying speed
Velocity vector is always along the tangent
There are two accelerations: one is toward the center (centripetal acceleration) and other is
along the tangent (tangential acceleration)
Angular acceleration is non-zero
Kinematics Equations in Circular motion
풂풕 = Tangential acceleration
Velocity after time t with initial velocity u and tangential acceleration 푎푡
푡
푣 = 푢 + ∫ 푎푡
0
푑푡
Tangential acceleration may be constant or a function of time.
Distance covered in time t with initial velocity u and tangential acceleration 푎푡
푡
푆 = ∫ (푢 + 푎푡
0
푡)푑푡
Velocity after covering distance s with initial velocity u and tangential acceleration 푎푡
Curvilinear Motion
푣2 − 푢2
2
푠
= ∫ 푎푡 푑푠
0
If the angle between the velocity and acceleration of a particle is other than 0 and 90, then the
motion of the particle is always along the curve.
Working Approach:
Resolve the initial velocity and acceleration along coordinate axis and represent them with
symbols : ux, uy, ax & ay
Note down the coordinate of the starting position of the particle
Use the equations of motion in 1-dimension separately for x-axis and y-axis

8. Radius of curvature
Suppose there is a particle moving along a curve.At any particular position of particle, we can
assume it to be present on a hypothetical circle touching the curve at that point on the inner side
of curve. The radius of this circle will have a fixed value and is known as radius of curvature ( )
of the path at that point.
Motion under Gravity
Projectile Motion
푅푐 =
푣2
푎푐
If the angle between the initial velocity of a particle and the acceleration due to gravity lies
between 90° to 180° (both exclusive), then the particle undergoes
projectile motion.
u = Initial velocity, g = Acceleration due to gravity, θ = Launching angle,
T = Time of flight
R = Range
Hm = Maximum height
Rm = Maximum range
Relative Motion
T =
2usinθ
g
R =
u2sin2θ
g
Hm =
u2sin2θ
2g
Rm =
u2sin2θ
2g
If the motion of a body is observed with respect to a moving reference frame, then it is called
relative motion. When we say velocity of A with respect to B has a particular value, then the sense
of statement is, that we assume reference frame to be at rest and subtract the velocity of
reference frame from absolute velocity of object and in that case, the considered motion of object
will give same result which we can get by considering separate absolute motion of both.

9. Types
Dependent relative motion
Independent relative motion
During relative motion of two particles, if their absolute motions are independent of each other,
then it is independent relative motion and if the absolute motions are dependent on each other,
then the motion is said to be dependent relative motion.
Angular velocity of line
It is defined as the ratio of the relative velocity perpendicular to the line to the length of the line.
Some Interesting Facts
Till the observer is at rest, displacement vector is independent of reference frame but position
vector depends on reference frame.
Average velocity and instantaneous velocity are in same direction when particle is moving in
straight line without reversing the direction.
Instantaneous speed is the magnitude of instantaneous velocity but average speed is not.
In a position-time graph, the point from which the concavity changes is the point of maximum or
minimum velocity.
If velocity and acceleration are perpendicular, then the motion is said to be uniform circular
motion.
At the instances of maximum or minimum speed, dv/dt is equal to zero.
The horizontal distance up to the maximum height in projectile motion is half times range.
Impact speed and impact angle in a projectile motion are equal to launching speed and launching
angle.
Speed at two points on the projectile on same horizontal level will be equal.
Sum of the launching angle and impact angle is equal to 90 for maximum range.
At the time of maximum range, the direction of initial velocity must be along the bisector of
angle b/w line of acceleration and line joining launching point and impact point

10. Minimum initial velocity for a projectile to cross a given point (x, y) is
√푔(푦 + √푥2 + 푦2)
The range is equal for two angles if the difference of these angles from range of maximum
angle is equal.
Radius of curvature of the projectile is minimum at the highest point of projectile.
Relative acceleration b/w two projectiles thrown at different speeds and angles is zero.
Is Kinematics an important part of IIT–JEE preparation?
‘Kinematics’ is an important part of mechanics which covers 20-25% of JEE physics paper every
year. You really need to master the concepts of this chapter to score high in IIT JEE.
What are the best books for the preparation of Kinematics?
Some of the books which are considered to be best for preparation of this section are:
Concepts of Physics by H.C. Verma
Problem in General Physics by I.E. Irodov
Fundamentals of Physics by Halliday, Resnick and Walker
Tips to study Kinematics for IIT JEE
You really need to understand this chapter if you want to boost your rank in JEE and other
engineering exams. This is the easiest chapter of mechanics but still due to
lack of clarity, people commit silly mistakes. Here are some tips which you
should follow to score high:
Don’t practice a single problem without hand and paper.
Important formulas and tips should be on your tips to save time in exams.
Practice as manyproblem as you can from various reference books and then try to solve previous
year JEE questions on Kinematics.
Make a different notebook in which you should write important concepts and formulas on daily
basis.
Don’t solve problems by seeing the answer; rather try to solve the problem using basic concepts.
Don’t read more than one book as referring too many books can lead to confusion. Always try to
stick to the one (Class notes/Resnick and Halliday/H.C. Verma)

11. While solving problems, read the questions carefully to know what all given in the question and
what is demanded.
Practical Application of Kinematics
Here are some of the beautiful examples of kinematics uses in real life
problems:
Kinematics is used extensively in calculating the real time distance,
velocity, and acceleration of various automobiles
The concepts of the kinematics are used in many sports like cricket, javelin throw, car racing,
etc.
The concept of relative velocity is used beautifully in determining the kinematics of the boat in
the river
Kinematics is an exciting area of computational mechanics which plays a central role in a variety
of fields and industrial applications.
Kinematics is used in the study of the paths of projectile like missiles and other arms.
Kinematics is used in determining the position and velocity of parachutes and upwardmoving
balloons
It is used in computing the velocity of various celestial bodies like planets, comets, etc.

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