Kinematics | Definition, Formula, Derivation, Problems

Kinematics is the study of motion of points, objects, and systems by examining their motion from a geometric perspective, without focusing on the forces that cause such movements or the physical characteristics of the objects involved. This study area uses algebra to create mathematical models that describe these motions, essentially treating it as the mathematics behind how things move.

Kinematics is a field of classical mechanics that deals with the motion of points, objects, and systems of objects. Kinematics is sometimes referred to as “motion geometry” by some professionals. Let’s have a look at the formula for kinematics.

In this article, we shall learn about kinematics, which is the study of motion, along with its formulas, derivation of kinematics formula, examples and others in detail.

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 [Tex]\bar{x}[/Tex], vertical [Tex]\bar{y}[/Tex], 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.

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

[Tex]{v}=v_{0}+at[/Tex]

[Tex]\Delta{x}=(\frac{v+v_{0}}{2})t[/Tex]

[Tex]\Delta{x}=v_{0}t+\frac{1}{2}at^{2}[/Tex]

[Tex]v^{2}=v_{0}^{2}+2a\Delta{x}[/Tex]

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

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-v₀ to 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 [Tex]\frac{1}{2}t(v-v_{0})     [/Tex] 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,

[Tex]\Delta{x}=v_{0}t+\frac{1}{2}t(v-v_{0})[/Tex]

[Tex]\Delta{x}=v_{0}t+\frac{1}{2}vt-\frac{1}{2}v_{0}t[/Tex]

[Tex]\Delta{x}=\frac{1}{2}vt+\frac{1}{2}v_{0}t[/Tex]

Finally, to obtain the second kinematic formula,

[Tex]\Delta{x}=(\frac{v+v_{0}}{2})t[/Tex]

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,

[Tex]\Delta{x}=v_{0}t+\frac{1}{2}at^{2}[/Tex]

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,

[Tex]v^{2}=v_{0}^{2}+2a\Delta{x}[/Tex]

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 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:

• 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:

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).

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

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:

Article Related to Kinematics:

• Difference Between Kinetics and Kinematics
• Kinematics of Rotational Motion
• Kinematic Equation of Motion: Time and Displacement
• Kinematic Viscosity Formula
• Difference Between Kinematic and Dynamic Viscosity

Question 1: In kinematics, what are the various variables?

Answer:

Distance, displacement, speed, velocity, acceleration, and jerk all affect the many variables in kinematics. Kinematics is not concerned with an object’s mass; rather, it is concerned with its motion. It’s completely descriptive and based on their observations, such as tossing a ball or operating a train.

Question 2: Give any Four Examples of Kinematics.

Answer:

Examples of Kinematics:

• A river’s flow.
• a stone flung from a great height.
• On a merry-go-round, children sit.
• swings of the pendulum.

Example 1: For the time span t = 7s, an automobile with a beginning velocity of zero accelerates uniformly at 16 m/s². Do you know how far it’s travelled?

Solution:

Given: t = 7s, v₀ = 0 m/s, a = 16 m/s²

Since,

[Tex]s=v_{0}t+\frac{1}{2}at^{2}[/Tex]

s = 0 × 7 + (1/2) × 16 × 7²

= (1/2) × 16 × 49

= 8 × 49

= 392 m

Example 2: A bicycle with initial velocity 2 experiences a uniform acceleration of 20 m/s² for the time interval 6s. Determine its Final velocity?  

Solution:

Given: v₀ = 2 m/s, a = 20 m/s², t = 6s

Since,

[Tex]v=v_{0}+at[/Tex]

v = 2 + 20 × 6

= 122 m/s

Example 3: Assume that the initial velocity is 0 and the final velocity is 5 for the time interval 4s then find its displacement?

Solution:

Given: v₀ = 0 m/s, v = 5 m/s, t = 4s

Since,

[Tex]\Delta{x}=(\frac{v+v_{0}}{2})t[/Tex]

Δx = (5+0) × 4

= 20 m

Example 4: Truck with initial velocity is zero, constant acceleration is 6 m/s^2, and time interval is 3s. Find the Final velocity?

Solution:

Given: v₀ = 0 m/s, a = 6 m/s², t = 3s

Since,

[Tex]v=v_{0}+at[/Tex]

= 0 + 6 × 3

= 18 m/s

Q1. A car travels at a constant speed of 60 km/h for 2 hours. How far does the car travel in this time?

Q2. A ball is thrown straight up with an initial velocity of 20 m/s from the ground. Assuming the acceleration due to gravity is -9.8 m/s², calculate the time it takes for the ball to reach its maximum height.

Q3. A stone is dropped from a cliff 80 meters high. How long does it take to hit the ground? Assume the acceleration due to gravity is 9.8 m/s².

Q4. A projectile is launched with an initial velocity of 50 m/s at an angle of 30° to the horizontal. Calculate the maximum height reached by the projectile. Ignore air resistance and use g = 9.8 m/s² for the acceleration due to gravity.

What is kinematics in physics?

Kinematics in physics is the branch that deals with the study of motion of objects without considering the forces that cause the motion. It focuses on the trajectories, velocities, and accelerations of moving objects.

What are the 4 types of kinematics?

The four types of kinematics typically refer to the different aspects or quantities that are studied within the branch, which include:

1. Displacement
2. Velocity
3. Acceleration
4. Time

What is kinematics 11th class?

In the 11th class physics curriculum, kinematics introduces students to the basic concepts of motion in a straight line, uniform and non-uniform motion, equations of motion, and graphical methods of describing motion. It lays the foundation for understanding the dynamics of motion (forces and laws of motion) which is usually covered subsequently

How to calculate speed?

Speed is calculated as the distance traveled divided by the time it takes to travel that distance. The formula for speed (v) is given by:

v = d/t​

What is the 5 formula of kinematics?

[Tex]{v}=v_{0}+at[/Tex]

[Tex]\Delta{x}=(\frac{v+v_{0}}{2})t[/Tex]

[Tex]\Delta{x}=v_{0}t+\frac{1}{2}at^{2}[/Tex]

[Tex]v^{2}=v_{0}^{2}+2a\Delta{x}[/Tex]

Why study kinematics?

kinematics is crucial for grasping how objects change their position and velocity over time, independently of the forces causing these changes. By comprehensively understanding object motion, we can subsequently delve into the application of forces on different objects.

What are the basic concepts of kinematics?

Kinematics, the study of motion, involves terms like distance, displacement, speed, velocity, and acceleration. These concepts describe an object’s position and how fast its motion changes.