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# Kinematics for JEE Advanced

The word Kinematics isn’t the friendliest word of all time. Kinematics JEE is even more frightening because it combines two words that can provoke fear in even the strongest of students! So let’s take a moment and break down this word and figure out what this scientific term means.

Kinematics of Machinery
By Fouad A. Saad

It is both fascinating and frustrating that scientists are always using words derived from different languages. The deeper you dive into the sciences, more you come across Greek and Latin. So the beginning of the word, ‘kinemat-‘ is Greek and means ‘motion’. The end of the word, ‘ics’ is Latin and means ‘the study of’. Therefore, kinematics is ‘the study of motion’. So, Kinematics is the study the motion of objects and groups of objects without considering the mass or the forces that caused the motion.

## An Incentive for Focusing on Kinematics

Kinematics is a part of mechanics and lies at the heart of physics. It fascinates me because if we know the present condition of an object, we can predict its future. We can get the answer to many questions like how far an object will travel or in which direction will it move or how quickly it can go from a dead stop to full speed.

In the JEE Main and JEE Advanced, the Kinematics plays an important role. Every year, more than ten questions are asked from mechanics. About 2 – 3 questions are solely based from this chapter, and the rest cannot be solved without a working knowledge of kinematics.

## Getting Started

Kinematics will become a nightmare if the basics are not clear to you. So, start from the basics. Don’t get overwhelmed at this point–just get started.

To start with the absolute basics, one can read NCERT thoroughly. The point of this blog is to highlight all the essential concepts, address common mistakes, address previous years’ questions related to Kinematics, and to make sure that you score better marks in the JEE Advanced. The concepts given here are in concise form and can be used for revision before the examination.

## List of Books

Resnick Halliday is an excellent book for theory. You can also buy Arihant books for Mechanics. They contain tons of problems and a lot of tips and tricks. Finally, to become unbeatable in Mechanics, solve Irodov. Trust me this book has a lot of challenging problems. After solving all the problems, you can easily crack the physics olympiad.

## Topics That Come under Kinematics

According to NCERT, the following topics come under Kinematics:

1. Rectilinear motion
2. Projectile motion
3. Circular Motion
4. Relative Motion

Now let’s see some theory.

## Rectilinear Motion

Rectilinear motion is the motion along a straight line or in one dimension. It deals with the kinematics of particle in one dimension. Now read and understand the following definitions:

1. Position
2. Displacement
3. Distance
4. Average velocity
5. Average speed
6. Instantaneous velocity
7. Instantaneous speed
8. Average acceleration
9. Instantaneous acceleration

One of the essential aspects of Kinematics is the study of graphs. Study the following graphs.

1. Position vs. Time
2. Velocity vs. Time
3. Speed vs. Time
4. Acceleration vs. Time
5. Velocity vs. Displacement

Important Points that you should keep a note of:

1. For uniformly accelerated motion, x-t graph is a parabola (opening upwards if a > 0 and opening downwards if a < 0). The slope of the tangent at any point of the parabola gives the velocity at that instant.
2. For uniformly accelerated motion, a v-t graph is a straight line whose slope gives the acceleration of the particle.
3. In general, the slope of the tangent in an x-t graph is velocity, and the slope of the tangent in a v-t graph is the acceleration.
4. The area under the a-t graph gives the change in velocity.
5. The area between the v-t graph gives the distance traveled by the particle if we take all areas as positive.
6. The area under v-t graph gives displacement if region below the x-axis is taken negatively.

Some important formulae for Uniformly Accelerated Motion:

• v = u + at
• s = ut + 0.5at2
• v2 = u2 + 2as

Be careful while applying these formulas. Maintaining proper sign convection is an absolute must. These formulae should be remembered by heart because they help in time management.

Some Formulas for Non-Uniformly Accelerated Motion:

• v = dx/dt
• a = dv/dt = v(dv/dx)

## Projectile Motion

Projectile motion is a form of motion experienced by an object or particle (a projectile) that is thrown near the Earth’s surface and moves along a curved path under the action of gravity.

Source:https://calculator.tutorvista.com/trajectory-calculator.html

### Some Formulae Related to Projectile Motion

Here vx is the velocity along the x-axis, ux is the initial velocity along the x-axis, vy is the velocity along the y-axis, uy is the initial velocity along the y-axis, g is the acceleration due to gravity, t is the time taken.

• Horizontal distance: x = vxt
• Horizontal velocity: vx = ux
• Vertical distance: y = uyt – 0.5gt2
• Vertical velocity: vy = uy – gt

u is the initial Velocity, sin θ is the component along the y-axis, and cos θ is the component along the x-axis.

• Time of flight: t = (2usinθ)/g
• Maximum height reached: H = (u2sin2θ)/g
• Horizontal range: R = (u2sin2θ)/g

Now let us derive the equation of trajectory of a projectile motion.
We know:

x = uxt = u cosθ t
y = uyt – 0.5gt2 = u sinθ t – 0.5gt2

Eliminating t from both the equations, we get:

### Important Points

1. The vertical component of velocity is zero when the particle moves horizontally, that is, at the highest point of the trajectory.
2. The vertical component of velocity is positive when the particle is moving up, and the vertical component of velocity is negative when the particle is coming down–if vertical upwards direction is taken as positive. Any direction upward or downward can be considered as positive, and if the downward direction is taken as positive, then the vertical component of velocity coming down is positive.

### General Result

1. For maximum range θ = 45°. Here Maximum height = Half of range.
2. We get the same range for two angles of projections θ and (90 – θ), but in both cases, maximum heights attained by the particles are different.
3. A Range can also be expressed as:

## Projectile Motion on an Inclined Plane

Till now we’ve seen ground to ground projections, but a projectile can be launched from an inclined plane also.
If we continue to use the natural axis system, it becomes tedious. So, to simplify our task we adopt a new axis system, that is–x-axis along the plane and the y-axis perpendicular to the plane.

### Some Formulas

In the case of projection on an inclined plane, remembering formulas can be a tough task. So, it’s advisable to solve problems by breaking down vectors along–and perpendicular to–the plane and apply formulas of rectilinear motion. One should not be afraid to play with vectors.

## Circular Motion

When a particle moves in a plane such that its distance from a fixed point remains constant, then its motion is called circular motion with respect to that fixed point.

Read and get a grasp of the following definitions:

1. Angular Position
2. Angular Displacement
3. Average Angular Velocity
4. Instantaneous Angular Velocity
5. Average Angular Acceleration
6. Instantaneous Angular Acceleration
7. Radial and tangential Acceleration
8. Centripetal and Centrifugal Force

### Some Formulas

• V=2πr/time
• ω=2π/T=2πf
• acentripetal= − 4π2r/T2
• acentripetal= −ω2r
• acentripetal= v2/r
• Fc= mv2/r

Here V is the tangential velocity, r is the radius of the circle, w is the angular velocity, a is the centripetal acceleration, and F is the centripetal force acting on the particle. Now let us see some relation among angular variables.

• ω = ω0 + αt
• θ= ω0t + ½αt2
• ω2 = ω02 + 2αθ

## Radius of Curvature

Any curved path can be assumed to be made of infinite circular arcs. The radius of curvature at a point is the radius of the circular arc at a particular point which fits the curve at that point.

Fc = mv2/R
=> R = mv2/Fc

Here R is the radius of the curvature. If the equation of trajectory of a particle is given, we can find the radius of curvature of the instantaneous circle by using this formula.

Only the theory won’t help, so solve the list of the following problems for better understanding.

1. Circular Motion in a horizontal plane
2. Circular Motion in a vertical plane
1. Conditions for oscillation when an object is attached with a
1. Light rod
2. String
2. Conditions for complete rotation when an object is attached with a
1. Light rod
2. String
3. Circular turning on roads
When vehicles go through turnings, they travel along a nearly circular arc. There must be some force which will produce the required centripetal acceleration. If the vehicles travel in a horizontal circular path, this resultant force is also horizontal. The necessary centripetal force is being provided to the vehicles by following three ways.

1. By friction only
2. By banking of roads only.
3. By friction and banking of roads both.
4. Centrifugal force and apparent weight on earth.

## Relative Motion

Relative motion of an object with respect to the observer is defined as the motion with which the object appears to move if the observer is considered to be at rest. The concept of relative motion is introduced to simplify solving problems. No new theory is taught. The only way to master the concept of relative motion is to solve as many problems as possible. Some important problem types are listed below:

1. Relative motion in a lift
2. The motion of a train moving on the equator
3. River problems(shortest time, shortest path, etc.)
4. Rain problems or wind airplane problems
5. Mother-daughter problems
6. Collision problems

## Tips and Tricks

1. Always draw free body diagrams. Even though it is too obvious, try to sketch the problem.
2. Always write down what is asked and what is given.
3. Solve zillions of problems.
4. To get the taste of the JEE pattern, solve all the previous years’ questions including the subjective questions.

## How to Outperform Yourself

The best thing would be first to classify the chapter into such profiles and identify which profile you are weak at. After doing this, you can start reading theory for the profile and simultaneously solve related questions. Put this philosophy to the test. You will be surprised at how much better you understand the concepts. And how much better you will do on your exams.

I hope this post will help you grasp a firm command over mechanics.

All the best!

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