Kinematics is the branch of physics that deals with the study of motion without considering the forces causing it.

Define Rest and Motion.

Rest: If position does not change with time, then it is at rest; Motion: If the position of a particle changes with time, then it is called in motion.

What is a Position vector?

A position vector specifies the location of a particle relative to a reference point, such as the origin. It is a vector that indicates the position of a specific point with respect to this reference. The direction of the position vector always points from the reference point (origin) toward the given point.

What is Displacement?

The shortest distance between the initial and final positions of the particle is called displacement

Kinematics: Definition, Equations & Applications in Physics

Type of Motion	Description	Examples
1-D Motion	Motion along a straight line, with position changing over time in one dimension.	Train on a straight track
2-D Motion	Motion in a plane, with position changing over time in two dimensions.	Earth revolves around the Sun
3-D Motion	Motion in space, with position changing over time in three dimensions.	Motion of an aeroplane

Concepts	Description	Key Points
Distance	The actual length of the path traveled by a particle between its initial and final positions.	Depends on the path taken, not just the initial and final positions. Scalar quantity. SI Unit: Meter (m), CGS Unit: Centimetre (cm). Dimension: $[M^{0} L^{1} T^{0}]$ Cannot be negative
Displacement	The shortest distance between the initial and final positions of the particle.	Vector quantity, direction from initial to final position. Depends only on initial and final positions, not the path taken. Can be positive, negative, or zero. Magnitude of Displacement ≤ Distance

East	West	North	South	Up	Down
$\hat{i}$	$- \hat{i}$	$\hat{j}$	$- \hat{j}$	$\hat{k}$	$- \hat{k}$

Also Read : Vectors in Physics

Types of Speed	Description	Example
Uniform Speed	A particle covers equal distances in equal intervals of time.	Speed remains constant. Example: A car moving at 20 meters every 5 seconds.
Non-uniform Speed	A particle covers unequal distances in equal intervals of time.	Speed varies over time. Example: A car moves 20m in 5s, 40m in the next 5s, and 10m in another 5s.
Average Speed	The ratio of total distance traveled to the total time taken over a given time interval.	Average speed =Total DistanceTotal Time Example: A car covers 60 meters in 5 seconds, average speed = 60m / 5s = 12 m/s.
Instantaneous Speed	The speed of a particle at a specific moment in time.	Speed at a particular instant. Example: The speed of a car at the 10-second mark.

Uniform Speed

Non-uniform Speed (Variable Speed)

Type of Velocity	Description	Key Points
Uniform Velocity	The velocity remains constant in both magnitude and direction.	Occurs when a particle moves in a straight line without changing direction.
Non-uniform Velocity	The velocity changes in magnitude, direction, or both.	Occurs when either speed or direction, or both, vary during motion.
Average Velocity	The ratio of displacement to the total time taken by the body.	$Average velocity = \frac{N e t D i s pl a ce m e n t}{T im e t ak e n}$ $V_{a vg} = \frac{Δ r}{Δ t} = \frac{r _{f} - r _{i}}{t _{f} - t _{i}}$ Its direction is along the displacement.

Type of Acceleration	Description	Key Points
Uniform Acceleration	Acceleration remains constant in both magnitude and direction during the motion of the particle.	The magnitude and direction of acceleration do not change.
Non-Uniform Acceleration	Acceleration changes in magnitude, direction, or both during the motion of the particle.	Either magnitude or direction, or both, of acceleration change.
Average Acceleration	The ratio of the total change in velocity to the total time taken by the particle.	$a_{a vg} = \frac{Δ v}{Δ t} = \frac{v _{f} - v _{i}}{t _{f} - t _{i}}$

$a = \frac{d v}{d t}$

$\int_{v}^{u} d v = \int_{0}^{t} a d t$

$v - u = a t$

$v = u + a t$

$v = \frac{d s}{d t}$

$\int_{0}^{s} d s = \int_{0}^{t} v d t$

$s = u t + \frac{1}{2} a t^{2}$

$a = v \frac{d v}{d s}$

$\int_{u}^{v} v d v = \int_{0}^{s} a d s$

$[\frac{v ^{2}}{2}]_{u}^{v} = a [s]_{0}^{s}$

$v^{2} = u^{2} + 2 a s$

Case-1

$θ = 0°$

$T an θ = 0°$

$v e l oc i t y = 0$

The body is at rest.

Case-2

$θ = co n s t an t$

$T an θ = co n s t an t$

$v e l oc i t y = co n s t an t$

the body is in uniform motion

Case-3

$θ > 90°$

$T an θ = n e g a t i v e$

velocity=-ve but constant uniform motion

Case-4

$θ$ is decreasing with time velocity is decreasing with time

Case-5

$θ$ is increasing with time velocity is increasing with time

Case-1

$θ = 0°$

$T an θ = 0°$

$a cce l er a t i o n = 0$

$v = co n s t an t$

uniform motion

Case-2

$θ = co n s t an t$

$T an θ = co n s t an t$

$a cce l er a t i o n = co n s t an t$

uniformly accelerated motion

Case-3

$θ > 90°$

$T an θ = n e g a t i v e$

$a cce l er a t i o n = - v e b u t co n s t an t$

uniform motion

Case-4

$θ$ is decreasing with time tan is decreasing with time acceleration is decreasing with time acceleration goes on decreasing with time but it is not retardation

Velocity-time graph when theta is decreasing

Case-5

$θ$ is increasing with time acceleration is increasing with time

Uniform or constant acceleration

Uniformly increasing acceleration.

$a \propto t^{0}$

Case-1 : Downward Projection

Case- 2: Upward Projection

$v = u + g t$

$h = u t + \frac{1}{2} g t^{2}$

$v^{2} = u^{2} + 2 g h$

$h_{n} = u + \frac{g}{2} (2 n - 1)$

$v_{max} = u^{2} + 2 g h$

$v = - u + g t$

$h = - u t + \frac{1}{2} g t^{2}$

$v^{2} = u^{2} + 2 g h$

$h_{n} = - u + \frac{g}{2} (2 n - 1)$

$H_{max} = H + \frac{u ^{2}}{2 g}$

$Distance travelled = H + \frac{u ^{2}}{g}$

For ascending motion(upward motion)

Time to go up, $t_{1} = \frac{2 h}{g + a}$

Velocity of projection, $v_{1} = 2 (g + a) h$

For descending motion (downward motion)

Time to go up, $t_{1} = \frac{2 h}{g - a}$

Velocity on reaching ground, $v_{1} = 2 (g - a) h$

Graphs of displacement with respect to time

Graph of velocity with respect to time

Graph of acceleration with respect to time

Graphs of displacement with respect to time

Graph of velocity with respect to time

Graph of acceleration with respect to time

Horizontal Motion	Vertical Motion
Initial velocity in horizontal direction $u_{x} = u cos θ$	Initial velocity in vertical direction $u_{y} = u s in θ$ Note: It is motion under the effect of gravity so that as particle moves upwards the magnitude of its vertical velocity decreases.
Acceleration along horizontal direction $a_{x} = 0$	Acceleration along vertical direction $a_{y} = - g$
At any instant horizontal velocity $v_{x} = u cos θ$	At any instant, vertical speed $v_{y} = u_{y} - g t = u s in θ - g t$
displacement along horizontal direction $x = (u cos θ) t$	displacement in vertical direction $y = u_{y} t - \frac{1}{2} g t^{2} = u s in θ - \frac{1}{2} g t^{2}$

Time of flight (T)	$T = \frac{2 u s in θ}{g}$
Time of ascent or Time of descent	$T = \frac{u s in θ}{g}$
Maximum height (H)	$H = \frac{u ^{2} s i n ^{2} θ}{2 g}$
Horizontal range or Range (R)	$R = \frac{u ^{2} s in 2 θ}{g}$
Maximum horizontal range (Rmax)	$R_{ma x} = \frac{u ^{2}}{g}$
Equation of Trajectory	$y = x t an θ [1 - \frac{x}{R}]$

Velocity at a General Point	$v = v_{x}^{2} + v_{y}^{2}$ $t an θ = \frac{g t}{u}$
Time of Flight	$T = \frac{2 h}{g}$
Horizontal Range	$R = u \frac{2 h}{g}$
Height	$h = \frac{1}{2} g T^{2}$

Case 1: Projection from a height at an angle θ above horizontal

Case 2: Projection from a height at an angle θ Below horizontal

Velocity after falling height h

$v_{y}^{2} = (- u s in θ)^{2} + 2 h g (v er t i c a l d i rec t i o n)$

$v_{x} = u_{x} = u cos θ$

$v = v_{x}^{2} + v_{y}^{2} = u^{2} + 2 g h$

Velocity after falling height h

$u_{y}^{2} = (u s in θ)^{2} + 2 h g (v er t i c a l d i rec t i o n)$

$v_{x} = u_{x} = u cos θ$

$v = v_{x}^{2} + v_{y}^{2} = u^{2} + 2 g h$

For same direction

For opposite directions

When two particles move in the same direction, their relative velocity equals the difference in their speeds.

Relative Displacement For same direction

$∣ v_{12} ∣ or ∣ v_{21} ∣ = v_{1} - v_{2}$

When two particles move in opposite directions, their relative velocity is the total of their speeds.

$∣ v_{12} ∣ or ∣ v_{21} ∣ = v_{1} - v_{2}$

If the swimming is in the direction of flow of water or along the downstream then

$v_{m} = v + v_{R}$

If the swimming is in the direction opposite to the flow of water or along the upstream then

$v_{m} = v - v_{R}$

If the man is crossing the river i.e. $v$ and $V_{R}$ are non collinear then use vector algebra.

$v_{m} = v + v_{R}$

Vectors in Physics	Conservation of Linear Momentum	Kinetic Energy
Newton’s Second Law of Motion	Electric Charges And Fields	Wave Motion
Moving Coil Galvanometer	Impulse	Carbon Resistor

Frequently Asked Questions

What is Kinematics?

Define Rest and Motion.

What is a Position vector?

What is Displacement?

Join ALLEN!

Frequently Asked Questions

What is Kinematics?

Define Rest and Motion.

What is a Position vector?

What is Displacement?

Join ALLEN!

Kinematics

1.0Basics of Kinematics

2.0Terminology Associated With Kinematics

3.0Speed

4.0Velocity

5.0Acceleration

6.0Velocity and Acceleration

7.0Equations of Motion

8.0Basic Graphs and Their Conversions

9.0Motion Under Gravity

Graphical Problems in Vertical Motion

10.0Motion In A Plane

11.0Relative Motion

12.0Rain-Man Concept

13.0River-Boat (or Man) Problem

14.0Sample Questions on Kinematics

Table of Contents