What is Kinematics?

Kinematics is the branch of engineering that examines the causes of movements and examines these movements in terms of direction, acceleration, and speed.

Contents

What is Kinematics?Kinematic Formulas Derivation of Kinematic Formulas Key Concept in Kinematics Position and Displacement Speed and Velocity Acceleration Kinematic Equations for Rotational Motion Motion Graph

Kinematics is concerned with the trajectories of points, lines, and other geometric objects to describe motion. Furthermore, it concentrates on deferential qualities such as velocity and acceleration. Astrophysics, mechanical engineering, robotics, and biomechanics all use kinematics extensively.

Kinematic formulas are a collection of equations that connect the five kinematic variables: Displacement (Δx), time interval (t), Initial Velocity (v₀), Final Velocity (v), Constant Acceleration (a).

Kinematic formulas are only accurate if the acceleration remains constant across the time frame in question; we must be careful not to apply them when the acceleration changes. The kinematic formulas also imply that all variables correspond to the same direction: horizontal $x ˉ$ , vertical $y ˉ$ , and so on.

Kinematics Definition

Kinematics is the study of motion in its simplest form. Kinematics is a branch of mathematics that deals with the motion of any object. The study of moving objects and their interactions is known as kinematics. Kinematics is also a branch of classical mechanics that describes and explains the motion of points, objects, and systems of bodies.

Kinematic Formulas

The kinematics formulas deal with displacement, velocity, time, and acceleration. In addition, the following are the four kinematic formulas:

Note that, One of the five kinematic variables in each kinematic formula is missing.

Derivation of Kinematic Formulas

Here is the derivation of the four-kinematics formula mentioned above:

Derivation of First Kinematic Formula

We have,

Acceleration = Velocity / Time

a = Δv / Δt

We can now use the definition of velocity change v-v₀to replace Δv.

a = (v-v₀)/ Δt

v = v₀ + aΔt

This becomes the first kinematic formula if we agree to just use t for Δt.

v = v_o + at

Derivation of Second Kinematic Formula

Displacement Δx can be found under any velocity graph. The object’s displacement Δx will be represented by the region beneath this velocity graph.

Δx is a total area, This region can be divided into a blue rectangle and a red triangle for ease of use.

The blue rectangle’s area is v₀t since its height is v₀ and its width is t. And The red triangle area is $2 1 t (v - v_{0})$ since its base is t and its height is v-v₀.

The sum of the areas of the blue rectangle and the red triangle will be the entire area,

$Δ x = v_{0} t + 2 1 t (v - v_{0})$

$Δ x = v_{0} t + 2 1 v t - 2 1 v_{0} t$

$Δ x = 2 1 v t + 2 1 v_{0} t$

Finally, to obtain the second kinematic formula,

$Δ x = (2 v + v ^{0}) t$

Derivation of Third Kinematic Formula

From Second Kinematic Formula,

Δx/t = (v+v₀)/2

put v = v₀ + at we get,

Δx/t = (v₀+at+v₀)/2

Δx/t = v₀+ at/2

Finally, to obtain the third kinematic formula,

$Δ x = v_{0} t + 2 1 a t^{2}$

Derivation of Fourth Kinematic Formula

From Second Kinematic Formula,

Δx = ((v+v0)/2)t

v=v₀+at …(From First Kinematic Formula)

t = (v-v₀)/a

Put the value of t in Second Kinematic Formula,

Δx = ((v+v₀)/2) × ((v-v₀)/a)

Δx = (v²+v₀²)/2a

We get Fourth Kinematic Formula by solving v²,

$v^{2} = v_{02} + 2 a Δ x$

Key Concept in Kinematics

Kinematics is a branch of mechanics that describes the motion of objects without considering the causes of this motion (i.e., forces). It involves the study of displacement, velocity, and acceleration of moving objects.

Displacement: Displacement is a vector quantity that refers to the change in position of an object. It has both magnitude and direction.

Δx = x_f – x_i

where,

x_f is the Final Position
x_i is the Initial Position

Velocity: Velocity is a vector quantity that denotes the rate of change of displacement with respect to time. It includes both speed and direction.

v_avg = Δx/Δt

where,

Δx is Displacement
Δt is the Time Interval

Acceleration: Acceleration is a vector quantity that represents the rate of change of velocity with respect to time.

a_avg = Δv/Δt

where,

Δv is Change in Velocity
Δt is the Time Interval

Relative Motion: Relative motion describes the motion of an object as observed from a particular frame of reference.

Position and Displacement

Position is a vector quantity that represents the location of an object in a given frame of reference. It is often described using coordinates in one, two, or three-dimensional space.

Displacement is a vector quantity that represents the change in position of an object. It is defined as the straight-line distance and direction from the initial position to the final position.

Differences Between Position and Displacement

Various differences between Position and Dispalcement are added in the table below:

Aspect	Position	Displacement
Definition	Location of an object in space	Change in position of an object
Type	Vector	Vector
Coordinates	Described by coordinates in 1D, 2D, or 3D space	Difference between initial and final position
Magnitude	Absolute value of coordinates	Straight-line distance between two positions
Direction	Not inherently directional (depends on reference)	Always directional (from initial to final)

Speed and Velocity

Speed is a scalar quantity that measures the rate at which an object covers distance. It only has magnitude and no direction. Speed is always positive or zero.

Velocity is a vector quantity that measures the rate of change of displacement with respect to time. It has both magnitude and direction. Velocity can be positive, negative, or zero.

Differences Between Speed and Velocity

Various differences between Speed and Velocity are added in the table below:

Aspect	Speed	Velocity
Quantity Type	Scalar	Vector
Components	Magnitude only	Magnitude and direction
Formula	v = d/t	v = Δx/t
Positive/Negative	Always positive or zero	Can be positive, negative, or zero
Direction	No direction	Specific direction
Example	60 km/hr (without direction)	60 km/hr north

Acceleration

Acceleration is the rate at which an object’s velocity changes with time. It indicates whether an object is speeding up, slowing down, or changing direction.

Average acceleration (a) is calculated by dividing the change in velocity (Δv) by the time interval (Δt) over which the change occurs:

a = Δv/Δt

where Δv =v_f – v_i (final velocity minus initial velocity).

SI unit of acceleration is meters per second squared (m/s²).

Kinematic Equations for Rotational Motion

Equation of motion for rotational motion are:

First Equation (Angular Velocity-Time Relation):

ω_f = ω_i + α_t

Second Equation (Angular Displacement-Time Relation):

θ = ω_it + 1/2αt²

Third Equation (Angular Velocity-Angular Displacement Relation):

ω_f² = ω_i² + 2αθ

Fourth Equation (Average Angular Velocity):

θ = (ω_i+ ω_f)t/2

where,

ω_f: Final Angular Velocity
ω_i: Initial Angular Velocity
α: Angular Acceleration
t: Time
θ: Angular displacement

Motion Graph

Various motion graphs are added below:

Displacement-Time Graph

Displacement-Time Graph is shown in the image added below:

Velocity-Time Graph

Velocity-Time Graph is shown in the image added below:

Acceleration-Time Graph

Acceleration-Time Graph is shown in the image added below:

What is Kinematics?