What is Kinematics? Definition, Formula, Derivation, Sample Problems

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Kinematics is the science of studying the motion of points, objects, and groups of objects while neglecting their causes. 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.

What is Kinematics?

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.

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

The kinematic formulas are a collection of equations that connect the five kinematic variables: Displacement $(\Delta{x})$ , time interval (t), Initial Velocity (v₀), Final Velocity (v), Constant Acceleration (a).

The 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 $\bar{x}$ , vertical $\bar{y}$ , and so on.

Kinematic Formulas

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

${v}=v_{0}+at$

$\Delta{x}=(\frac{v+v_{0}}{2})t$

$\Delta{x}=v_{0}t+\frac{1}{2}at^{2}$

$v^{2}=v_{0}^{2}+2a\Delta{x}$

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

or

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.

${v}=v_{0}+at$

Derivation of Second Kinematic Formula

The 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 $\frac{1}{2}t(v-v_{0})$ 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,

$\Delta{x}=v_{0}t+\frac{1}{2}t(v-v_{0})$

$\Delta{x}=v_{0}t+\frac{1}{2}vt-\frac{1}{2}v_{0}t$

$\Delta{x}=\frac{1}{2}vt+\frac{1}{2}v_{0}t$

Finally, to obtain the second kinematic formula,

$\Delta{x}=(\frac{v+v_{0}}{2})t$

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,

$\Delta{x}=v_{0}t+\frac{1}{2}at^{2}$

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,

$v^{2}=v_{0}^{2}+2a\Delta{x}$

Sample Questions

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.

Question 3: 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?

Answer:

Given: t = 7s, v₀= 0 m/s, a = 16 m/s²

Since,

$s=v_{0}t+\frac{1}{2}at^{2}$

s = 0 × 7 + (1/2) × 16 × 7²

= (1/2) × 16 × 49

= 8 × 49

= 392 m

Question 4: A bicycle with initial velocity 2 experiences a uniform acceleration of 20 m/s² for the time interval 6s. Determine its Final velocity?

Answer:

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

Since,

$v=v_{0}+at$

v = 2 + 20 × 6

= 122 m/s

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

Answer:

Given: v₀ = 0 m/s, v = 5 m/s, t = 4s

Since,

$\Delta{x}=(\frac{v+v_{0}}{2})t$

Δx = (5+0) × 4

= 20 m

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

Answer:

Given: v₀ = 0 m/s, a = 6 m/s², t = 3s

Since,

$v=v_{0}+at$

= 0 + 6 × 3

= 18 m/s

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Last Updated : 31 Aug, 2022

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