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.

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 𝑟 be the position vector of a particle at any instant t. Then instantaneous velocity �⃗� is defined

as

Note: Instantaneous speed is magnitude of instantaneous velocity.

Average Velocity

Average Speed

Average Speed = Distance/Time

Note: Average velocity is in the direction of displacement vector.

�⃗� = 𝑑𝑟

𝑑𝑡

𝑣𝑎𝑣⃗⃗⃗⃗⃗⃗⃗ = ∆𝑟

∆𝑡

Acceleration

Instantaneous Acceleration

It is defined as rate of change of velocity vector �⃗�.

Average Acceleration

Motion in 1-dimension

Points to remember:

Velocity (�⃗�) is always along a fixed line.

Acceleration (�⃗�) is either parallel (accelerated motion) or anti-parallel (retarded motion) to

velocity (�⃗�).

Important formulas

Uniform acceleration

Velocity of particle after time t with initial velocity �⃗⃗� and uniform acceleration �⃗� is

Displacement covered by particle in time t with initial velocity �⃗⃗� and uniform acceleration �⃗� is

Velocity of particle after covering displacement 𝑠 with initial velocity �⃗⃗� and uniform acceleration �⃗�

is

Displacement covered in a 𝑛𝑡ℎ second with initial velocity �⃗⃗� and uniform acceleration �⃗� is

�⃗� = 𝑑�⃗�

𝑑𝑡

𝑎𝑎𝑣⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ∆�⃗�

∆𝑡

�⃗� = �⃗⃗� + �⃗�𝑡

𝑆 = �⃗⃗�𝑡 + 1

2�⃗�𝑡2

�⃗�2 − �⃗⃗�2 = 2�⃗�𝑠

𝑆𝑛 = �⃗⃗� +1

2�⃗�(2𝑛 − 1)

Non – uniform acceleration

Calculus Approach

Acceleration as a function of time

𝒂 = 𝒇(𝒕)

𝒅𝒗

𝒅𝒕= 𝒇(𝒕)

∫ 𝒅𝒗 = ∫ 𝒇(𝒕)𝒅𝒕

From here, we will get velocity – time equation

𝒗 = 𝒇(𝒕)

𝒅𝒙

𝒅𝒕= 𝒇(𝒕)

∫ 𝒅𝒙 = ∫ 𝒇(𝒕)𝒅𝒕

We will get the position – time equation.

Acceleration as a function of velocity (v⃗⃗)

𝒂 = 𝒇(𝒗)

𝒅𝒗

𝒅𝒕= 𝒇(𝒗)

∫𝒅𝒗

𝒇(𝒗)= ∫ 𝒅𝒕

We will get velocity – time equation from here.

Similarly, we will get position – time equation by putting

𝒗 = 𝒅𝒙

𝒅𝒕

Acceleration as a function of position

𝒂 = 𝒇(𝒙)

𝒗𝒅𝒗

𝒅𝒙= 𝒇(𝒙)

∫ 𝒗𝒅𝒗 = ∫ 𝒇(𝒙)𝒅𝒙

We will get velocity-position equation from here

And by putting

𝒗 = 𝒅𝒙

𝒅𝒕

We will get position-time equation.

Position as a function of time

𝒙 = 𝒇(𝒕)

𝒗 = 𝒅𝒙

𝒅𝒕

𝒂 = 𝒅𝒗

𝒅𝒕

Velocity as a function of position

𝒗 = 𝒇(𝒙)

𝒅𝒙

𝒅𝒕= 𝒇(𝒙)

∫𝒅𝒙

𝒇(𝒙)= ∫ 𝒅𝒕

We will get position-time equation from here and then repeat the working of previous case.

Velocity as a function of time

𝒗 = 𝒇(𝒕)

𝒅𝒙

𝒅𝒕= 𝒇(𝒕)

∫ 𝒅𝒙 = ∫ 𝒇(𝒕)𝒅𝒕

We will get position-time equation from here

Acceleration-time equation

𝒂 = 𝒅𝒗

𝒅𝒕

Motion in 2-dimension

Circular Motion

Angular Displacement (∆𝜃): Change in angular position of a particle.

∆𝜽 = 𝜽𝒇 − 𝜽𝒊

Angular Velocity (𝜔):

𝝎 =𝒅𝜽

𝒅𝒕 𝝎𝒂𝒗𝒆. =

𝜽𝒇 − 𝜽𝒊

𝒕

Angular Acceleration (α):

𝜶 = 𝒅𝝎

𝒅𝒕 𝜶𝒂𝒗𝒆. =

𝝎𝒇 − 𝝎𝒊

𝒕

Relation b/w linear and angular variables

𝒍 = 𝒓𝜽

�⃗⃗⃗� = �⃗⃗⃗⃗� × �⃗⃗�

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 (𝑎𝑐⃗⃗⃗⃗⃗)

Angular velocity is constant and angular acceleration is zero

Let the position vector be 𝑟 and 𝜃 be angular position w.r.t. to some reference line, then

Velocity, v = dr

dt

Speed = Magnitude of v = rdθ

dt

Acceleration, a = dv

dt

Magnitude of a = vdθ

dt

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

𝒂𝒕 = Tangential acceleration

Velocity after time t with initial velocity u and tangential acceleration 𝑎𝑡

Tangential acceleration may be constant or a function of time.

Distance covered in time t with initial velocity u and tangential acceleration 𝑎𝑡

Velocity after covering distance s with initial velocity u and tangential acceleration 𝑎𝑡

Curvilinear Motion

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

𝑣 = 𝑢 + ∫ 𝑎𝑡

𝑡

0

𝑑𝑡

𝑆 = ∫ (𝑢 + 𝑎𝑡

𝑡

0

𝑡)𝑑𝑡

𝑣2 − 𝑢2

2= ∫ 𝑎𝑡𝑑𝑠

𝑠

0

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

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

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.

𝑅𝑐 = 𝑣2

𝑎𝑐

T =2usinθ

g

R =u2sin2θ

g

Hm =u2sin2θ

2g

Rm =u2sin2θ

2g

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

Minimum initial velocity for a projectile to cross a given point (x, y) is

√𝑔(𝑦 + √𝑥2 + 𝑦2)

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)

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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