JEE 2022 | Class 12

JEE Important Chapter - Kinematics

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What is Kinematics?

What is Kinematics?

Kinematics is an important topic for JEE exams. It mainly consists of one-dimensional and two-dimensional motion, and the questions asked in the examination are based on their applications. Compared to other concepts, this section will become easier with adequate practice and conceptual understanding.


In kinematics, first, we will learn about motion along a straight line and discuss different physical quantities like distance, displacement, velocity, etc., and their formulas. Different types of motions are also discussed in this section. Problems related to the three kinematical equations of motion for uniformly accelerated motion are very important. We will also see how we can draw a conclusion about the motion by evaluating and finding required quantities from various graphs.


Coming to motion along a plane, properties of vectors and the law of vector addition like triangular law and parallelogram law of vector additions, scalar and vector product of vectors are important. The problems with projectile motion on a horizontal and inclined plane are frequently asked in JEE exam.


Let us see what is kinematics and the important concepts and formulas needed to score a full marks for this topic in JEE Main with the help of some solved kinematics examples.


Kinematics - Important Topics 

  • Motion in a Straight Line

  • Kinematics Equations

  • Motion under gravity

  • Graphs for all types of motion

  • Motion in a Plane 

  • Projectile Motion

  • Resolution of Vector Rectangular Components

  • Vector Addition

  • Circular Motion


Important Concepts of Kinematics

Concepts

Key Points 

1. Motion along a straight line

  • Kinematics is a branch of physics that deals with the study of the motion of a point body, the system of bodies without considering the cause of the motion.

  • The motion is in which an object moves in a straight line and only one coordinate of the object changes with time.

  • Motion along a straight line can be represented as motion along with an x-axis.

  • Eg: The motion of a car along a straight line, a ball under free fall.

  • Instantaneous velocity is the velocity of an object at a particular instant.

$v=\lim_{\Delta t \rightarrow 0} \dfrac{\Delta x}{\Delta t} = \dfrac {dx}{dt}$

  • Instantaneous acceleration is the rate of change in velocity at a particular instant.

$a=\lim_{\Delta t \rightarrow 0} \dfrac{\Delta v}{\Delta t} = \dfrac {dv}{dt}$


2. Kinematics  formulas

  • Kinematics formulas are valid only for uniformly accelerated motion.

  • $v=u+at$

  • $s=ut+\dfrac{1}{2}at^{2}$

  • $v^2=u^2+2as$

Where u is the initial velocity, v is the final velocity, a is the acceleration and t is the time taken.

3. Graphs for different types of motion

For uniform motion, velocity remains constant. Velocity is not constant under non-uniform motion. For uniform velocity motion, acceleration remains constant.


  • Displacement-time graph

The slope of a displacement time graph gives velocity.

Displacement-time graph for different types of motion

  • Velocity-time graph

The slope of a velocity-time graph gives the displacement covered by the body.

velocity-time graph for different types of motion



  • Acceleration-time graph

The area under the curve of the acceleration time graph gives the change in velocity.

acceleration time graph for different types of motion


4. Motion under Gravity

  • If a body is thrown vertically upward or released from a height, it moves in a straight line under the influence of gravity.

  • A motion under gravity is a uniformly accelerated motion. Therefore, we can apply kinematics formulas.

  • While applying kinematics formulas, each term has to be substituted with a proper sign.

  • You can take an upward or downward direction as a positive sign. But remember that the opposite direction should be given a negative sign.

  • In the diagram below, a ball comes down after it is thrown upward. If the upward direction is taken as +ve, then initial velocity is +ve(upward direction) and final velocity and acceleration are -ve due to gravity is -ve because their direction is in the downward direction.

Direction of velocity and acceleration due to gravity when

5. Motion along a plane

  • Motion along a plane is also called two-dimensional motion.

  • For a body in two-dimensional motion,  its motion can be represented along with an XY plane.

  • Examples: circular motion, projectile motion

6. Vector addition

  • Parallelogram law of vector addition

If two vectors can be represented as two adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the resultant vector of the two vectors.

diagrammatic representation of Parallelogram law of vector addition


7. Resolution of a vector into rectangular components

  • A vector can be resolved into rectangular components, along its X- and Y-axis.

Resolution of vectors into rectangular components


  • $v_x=v\cos\theta$

$v_y=v\sin\theta$

8. Projectile motion

  • When an object is thrown at an angle with the horizontal, it follows a parabolic path under the influence of gravity called Projectile motion.

velocity at different points on a projectile motion


  • Range is the maximum horizontal distance covered during the projectile motion.

$R=\dfrac{u^2\sin (2\theta)}{g}$

  • Time of flight is the total time taken by the body to reach back to the ground.

$T=\dfrac{2u\sin (\theta)}{g}$

  • Maximum height of the projectile motion is given by,

$R=\dfrac{u^2\sin^2 (\theta)}{2g}$

Where u is the velocity at which the body is projected, θ is the angle of projection and g is the acceleration due to gravity. 


9. Circular motion

  • A body is said to be in a circular motion if it undergoes a circular path.

a particle moving in circular path


  • In a uniform circular motion, speed remains constant.

  • The relation between linear velocity (v) and angular velocity (⍵) of a particle moving in a circle of radius r is,

$v=r\omega$

  • Angular acceleration is the rate of change of angular velocity with respect to time.

$\alpha=\dfrac{d\omega}{dt}=r\omega^2$

  • The relationship between linear acceleration (a) and angular acceleration (α) is

$a=r\alpha$


List of Important Formulae

Sl. No.

Name of the Concept

Formula

1.

Acceleration formula for any type of motion

$a=\dfrac{dv}{dt}$

$v=\int_{t_1}^{t_2}a.dt$

2.

Addition and subtraction of two vectors

$|\vec A  +\vec B|=\sqrt{A^2+B^2+2AB\cos\theta}$

Where A and B are the magnitude of the two vectors at an angle θ.

3.

Subtraction of two vectors

$|\vec A  -\vec B|=\sqrt{A^2+B^2-2AB\cos\theta}$

Where A and B are the magnitude of the two vectors at an angle θ.

4.

Vector product of two vectors

$\vec A \times\vec B=AB\sin (\theta) \hat n$

Where A and B are the magnitude of two vectors at angle θ between them and $\hat n$ is the unit vector perpendicular to the plane containing $\vec A$ and {\vec B}

5.

Scalar product of two vectors

$\vec A \cdot\vec B=AB\cos (\theta) $

Where A and B are the magnitude of two vectors at angle θ between them

6.

The average velocity of the body in a journey that covers two equal distances with two different speeds


$v_{avg}=\dfrac{2uv}{u+v}$

Where u and v are the two different speeds by which the body covers two equal distances.

7.

The displacement covered in nth second of a uniformly accelerated body

$S_{n}=u+\dfrac{1}{2}a(2n-1)$

Where u is the initial velocity, a is the acceleration and Sn is the displacement covered by the body in the nth second.

8.

The motion of a body is projected from the top of a tower

The motion of a body is projected from the top of a tower


The trajectory of the body is given as

$y=\dfrac{gx^2}{2u^2}$

Where y is the vertical displacement and x is the horizontal displacement. 


Solved Examples 

1. A balloon carrying a stone rises from rest on the ground with an acceleration of 10 m/s2 in the upward direction. After 5 sec, the stone is released and ultimately reaches the ground. Find the maximum height attained by the height.

Ans:  Here, we will take the upward direction as positive and the downward direction as negative. Given the balloon rises up from rest. Therefore,


The initial velocity of the balloon, u1 = 0 m/s

Acceleration of the ball, a = 10 m/s2


First of all, let us find the height reached by the balloon in 5 seconds before the stone is released using the equation of motion.

$s=ut+\dfrac{1}{2}at^{2}$

Substitute the value of each term in the equation to calculate the height reached by the balloon.

$S=0\times 5+\dfrac{1}{2}\times 10\times 5^{2}$

$S= 125 m$

Now, even after the stone is released, it will continue to move upward for some time because it has some velocity in the upward direction. Then, the initial velocity of the stone is equal to the final velocity of the balloon after 5 sec. So, let us calculate the final velocity of the balloon using the formula given below:

$v=u+at$

Substitute the values and calculate the final velocity.

$v=0+10\times 5$

$v= 50 m/s$

Now, calculate the distance covered by the stone to reach the maximum height where the final velocity is zero by using the formula given below. (Take acceleration due to gravity as -10 m/s2 since it is in the downward direction.)

$v^2=u^2+2as$

$0^2=50^2+2\times (-10)\times$

$20s=2500$

$s=125 m$


Therefore, the maximum height attained by the stone is the sum of the distance travelled by the balloon in 5 sec and the distance travelled by the stone after it is released, i.e, 125 m + 125 m = 250 m


Key Point: The velocity of any body thrown upward will be zero when it reaches its highest point but the acceleration will be equal to the acceleration due to gravity acting downwards.


2. An aeroplane moving at a speed of 180 km/hr drops a food packet while flying at a height of 500 m. Find the horizontal range?

Ans: We will consider the motion along with vertical and horizontal directions separately to apply equations of motion. 


First, find the time taken by the packet to reach the ground considering motion along the vertical direction.

motion of food packet under gravity


The vertical displacement of the packet, Sy = h = 500 m


Initial velocity along vertical direction, uy = 0 m/s


Acceleration of the packet along vertical direction = g = 10 m/s2

$s_y=u_yt+\dfrac{1}{2}a_yt^{2}$

$S_y=u_yt+\dfrac{1}{2}a_yt^{2}$

$500 =0\times t+\dfrac{1}{2}\times 10t^{2}$

$1000=10t^{2}$

$t^2=100$

$t=10s$

Now consider the motion along horizontal direction and calculate the horizontal range using the equation of motion.

Initial velocity along the horizontal direction, ux = 180 km/hr = $180\times \dfrac{5}{18}$ = 50 m/s

Acceleration along the horizontal direction, ax = 0 m/s2

Time taken to reach the ground, t = 10s

$S_x=u_xt+\dfrac{1}{2}a_xt^{2}$

$S=50\times10+\dfrac{1}{2}\times 0\times 10^{2}$

$S_x= 500 m$

Key Point:  Whenever dealing with the problems related to the two-dimensional motion of a body under gravity, consider the motion along with vertical and horizontal directions separately to apply equations of motion.


Previous Year Questions from JEE Exam

1. Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by bullets fired by two guns on the ground is… (JEE 2019)

Ans: For a projectile motion, maximum range is attained at an angle of 45o. The area covered by the bullets will be equal to the area of the circle with a radius equal to the maximum range of each bullet.


Bullets fired by gun A:


Let RA be the maximum range of the bullets fired by gun A at an angle of  45o.


The velocity of the bullets fired by gun A, uA = 1 km/s


The maximum range of the bullet from gun A is given by

$R_A=\dfrac{u_A^2\sin(2\theta)}{g}$

$R_A=\dfrac{u_A^2\sin(2\times45)}{g}$

$R_A=\dfrac{u_A^2}{g}$

Maximum area covered by the bullets from gun A is

$Area_A=\pi R_A^2$

$Area_A=\pi\left( \dfrac{u_A^4}{g^2}\right) $


Similarly,


The velocity of the bullets fired by gun B, uB = 2 km/s 


The maximum range of the bullet from gun B is given by

$R_B=\dfrac{u_B^2}{g}$

Maximum area covered by the bullets from gun B is

$Area_B=\pi R_B^2$

$Area_B=\pi\left( \dfrac{u_B^4}{g^2}\right) $

Then the ratio of two area covered by the bullets fired by gun A and gun B is,

$\dfrac{Area_A}{Area_B}=\dfrac{\pi\left( \dfrac{u_A^4}{g^2}\right)}{\pi\left( \dfrac{u_B^4}{g^2}\right)}$

$\dfrac{Area_A}{Area_B}=\dfrac{u_A^4}{u_B^4}=\dfrac{1^4}{2^4}=\dfrac{1}{16} $ Therefore, the ratio of area covered by the bullets A and B is 1:16.


Key Point:  The maximum range is attained by a given projectile for a given velocity at an angle of 450


2. A stone is dropped from the top of a building. When it crosses a point 5m below the top, another stone starts to fall from a point 25m below the top, both stones reach the bottom of the building simultaneously. The height of the building is: (Take g = 10 m/s2) (JEE 2021 Feb)

Position of the stone at different points


Ans: First, we have to  calculate the velocity of the first stone (u1) at a height 5 m below the top of the building using the formula given below. 

$u_1=\sqrt {2gh}$

$u_1=\sqrt {2\times 10\times 5}=10m/s$

Now, apply the 2nd equation of motion on the first stone at a point 5 m below from the top till it reaches the ground. Then the displacement covered by the first stone is 20+h. 

$20+h=u_1t+\dfrac{1}{2}gt^{2}$

$20+h=10t+\dfrac{1}{2}gt^{2}$.....(1)

For the second stone, the displacement covered to reach the ground is h and the time taken is t. Using the 2nd equation of motion,

$20+h=10t+\dfrac{1}{2}gt^{2}$.....(2)

Put equation (2) in equation (1) to calculate the time t

$20+\dfrac{1}{2}gt^{2}=10t+\dfrac{1}{2}gt^{2}$

t=2s

Now, put t=2s in equation (2) to calculate h


$h=\dfrac{1}{2}\times 10\times 2^{2}=20 m$


The total height of the building = h + 25= 20 + 25 = 45 m


Key Point:  Whenever an object is dropped or released, the initial velocity must be taken as zero.


Practice Questions

  1. A body is projected at 600 with the ground. It covers a horizontal distance of 100 m. If the same body is projected at 600 with vertical with the same velocity, what will be the new range? (Ans: 100 m)

  2. A car moving with a speed of 50 km/hr, can be stopped by brakes after at least 6m. If the same car is moving at a speed of 100 km/hr, what will be the minimum stopping distance? (Ans: 24 m)


Conclusion

So here we discussed important concepts and formulas related to Kinematics from the JEE point of view. Students must make sure that they do not miss any of the above important topics to obtain a good score in the JEE Main exam.

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FAQs on JEE Important Chapter - Kinematics

FAQ

1. How many questions are asked from Kinematics in JEE Main?

Mostly, 1-2 questions from this chapter are asked in JEE Main every year which corresponds to around 8 marks in the JEE Main exam.

2. What is the importance of kinematics in the JEE exam?

Compared to other chapters, kinematics concepts will become easier for you once you start practicing more and more questions related to it. You can easily solve all the types of kinematics problems asked from this chapter once you learn all the topics covered in the chapter conceptually. Besides, the concepts discussed in this chapter are linked to problems from other chapters too.