Kinematics Preparation Tips for IIT JEE | askIITians

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 vector of the particle and

be the initial position

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 đ?‘&#x;⃗ be the position vector of a particle at any instant t. Then instantaneous velocity đ?‘Łâƒ— is defined as đ?‘Łâƒ— =

đ?‘‘đ?‘&#x;⃗ đ?‘‘đ?‘Ą

Note: Instantaneous speed is magnitude of instantaneous velocity.

Average Velocity đ?‘Łđ?‘Žđ?‘Ł = ⃗⃗⃗⃗⃗⃗⃗

∆đ?‘&#x;⃗ ∆đ?‘Ą

Average Speed Average Speed = Distance/Time Note: Average velocity is in the direction of displacement vector.

Acceleration Instantaneous Acceleration It is defined as rate of change of velocity vector đ?‘Łâƒ—. đ?‘Žâƒ— =

đ?‘‘đ?‘Łâƒ— đ?‘‘đ?‘Ą

đ?‘Žđ?‘Žđ?‘Ł = ⃗⃗⃗⃗⃗⃗⃗

∆đ?‘Łâƒ— ∆đ?‘Ą

Average Acceleration

Motion in 1-dimension Points to remember: Velocity (đ?‘Łâƒ—) is always along a fixed line.

Acceleration (đ?‘Žâƒ—) is either parallel (accelerated motion) or anti-parallel (retarded motion) to velocity (đ?‘Łâƒ—).

Important formulas Uniform acceleration Velocity of particle after time t with initial velocity đ?‘˘ ⃗⃗ and uniform acceleration đ?‘Žâƒ— is đ?‘Łâƒ— = đ?‘˘ ⃗⃗ + đ?‘Žâƒ—đ?‘Ą Displacement covered by particle in time t with initial velocity đ?‘˘ ⃗⃗ and uniform acceleration đ?‘Žâƒ— is đ?‘†âƒ— = đ?‘˘ ⃗⃗đ?‘Ą +

1 2 đ?‘Žâƒ—đ?‘Ą 2

Velocity of particle after covering displacement đ?‘ ⃗ with initial velocity đ?‘˘ ⃗⃗ and uniform acceleration đ?‘Žâƒ— is

đ?‘Łâƒ— 2 − đ?‘˘ ⃗⃗2 = 2đ?‘Žâƒ—đ?‘ ⃗

Displacement covered in a đ?‘›đ?‘Ąâ„Ž second with initial velocity đ?‘˘ ⃗⃗ and uniform acceleration đ?‘Žâƒ— is 1 đ?‘†âƒ—đ?‘› = đ?‘˘ ⃗⃗ + đ?‘Žâƒ—(2đ?‘› − 1) 2

Non – uniform acceleration Calculus Approach Acceleration as a function of time đ?’‚ = đ?’‡(đ?’•) đ?’…đ?’— = đ?’‡(đ?’•) đ?’…đ?’• âˆŤ đ?’…đ?’— = âˆŤ đ?’‡(đ?’•)đ?’…đ?’• From here, we will get velocity – time equation đ?’— = đ?’‡(đ?’•) đ?’…đ?’™ = đ?’‡(đ?’•) đ?’…đ?’• âˆŤ đ?’…đ?’™ = âˆŤ đ?’‡(đ?’•)đ?’…đ?’• We will get the position – time equation.

Acceleration as a function of velocity (v ⃗⃗) đ?’‚ = đ?’‡(đ?’—) đ?’…đ?’— = đ?’‡(đ?’—) đ?’…đ?’• âˆŤ

đ?’…đ?’— = âˆŤ đ?’…đ?’• đ?’‡(đ?’—)

We will get velocity – time equation from here.

Similarly, we will get position – time equation by putting �=

đ?’…đ?’™ đ?’…đ?’•

Acceleration as a function of position đ?’‚ = đ?’‡(đ?’™) đ?’—

đ?’…đ?’— = đ?’‡(đ?’™) đ?’…đ?’™

âˆŤ đ?’—đ?’…đ?’— = âˆŤ đ?’‡(đ?’™)đ?’…đ?’™

We will get velocity-position equation from here And by putting đ?’—=

đ?’…đ?’™ đ?’…đ?’•

We will get position-time equation.

Position as a function of time đ?’™ = đ?’‡(đ?’•) đ?’—=

đ?’…đ?’™ đ?’…đ?’•

đ?’‚=

đ?’…đ?’— đ?’…đ?’•

Velocity as a function of position đ?’— = đ?’‡(đ?’™) đ?’…đ?’™ = đ?’‡(đ?’™) đ?’…đ?’• âˆŤ

đ?’…đ?’™ = âˆŤ đ?’…đ?’• đ?’‡(đ?’™)

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

Velocity as a function of time đ?’— = đ?’‡(đ?’•) đ?’…đ?’™ = đ?’‡(đ?’•) đ?’…đ?’• âˆŤ đ?’…đ?’™ = âˆŤ đ?’‡(đ?’•)đ?’…đ?’• We will get position-time equation from here Acceleration-time equation đ?’‚=

đ?’…đ?’— đ?’…đ?’•

Motion in 2-dimension Circular Motion Angular Displacement (∆đ?œƒ): Change in angular position of a particle. ∆đ?œ˝ = đ?œ˝đ?’‡ − đ?œ˝đ?’Š

Angular Velocity (đ?œ”): đ??Ž=

đ?’…đ?œ˝ đ?’…đ?’•

đ??Žđ?’‚đ?’—đ?’†. =

đ?œ˝đ?’‡ − đ?œ˝đ?’Š đ?’•

Angular Acceleration (Îą): đ?œś=

đ?’…đ??Ž đ?’…đ?’•

đ?œśđ?’‚đ?’—đ?’†. =

đ??Žđ?’‡ − đ??Žđ?’Š đ?’•

Relation b/w linear and angular variables đ?’? = đ?’“đ?œ˝ ⃗⃗ = đ??Ž ⃗⃗⃗⃗ Ă— đ?’“ ⃗⃗ đ?’— 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 (đ?‘Ž ⃗⃗⃗⃗⃗) đ?‘?

Angular velocity is constant and angular acceleration is zero Let the position vector be đ?‘&#x;⃗ and đ?œƒ be angular position w.r.t. to some reference line, then Velocity, v =

dr dt

Speed = Magnitude of v = r Acceleration, a =

dv dt

Magnitude of a = v

dθ dt

dθ 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 đ?’‚đ?’• = Tangential acceleration Velocity after time t with initial velocity u and tangential acceleration đ?‘Žđ?‘Ą đ?‘Ą

đ?‘Ł = đ?‘˘ + âˆŤ đ?‘Žđ?‘Ą đ?‘‘đ?‘Ą 0

Tangential acceleration may be constant or a function of time.

Distance covered in time t with initial velocity u and tangential acceleration đ?‘Žđ?‘Ą đ?‘Ą

đ?‘† = âˆŤ (đ?‘˘ + đ?‘Žđ?‘Ą đ?‘Ą)đ?‘‘đ?‘Ą 0

Velocity after covering distance s with initial velocity u and tangential acceleration đ?‘Žđ?‘Ą đ?‘ đ?‘Ł 2 − đ?‘˘2 = âˆŤ đ?‘Žđ?‘Ą đ?‘‘đ?‘ 2 0

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 , a x & a y 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

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. đ?‘…đ?‘? =

đ?‘Ł2 đ?‘Žđ?‘?

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

T=

2usinθ g

R = Range

R=

u2 sin2θ g

Hm = Maximum height

Rm = Maximum range

Hm =

u2 sin2 θ 2g

Rm =

u2 sin2 θ 2g

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.

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 √đ?‘”(đ?‘Ś + √đ?‘Ľ 2 + đ?‘Ś 2 ) 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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