Differentiation and Integration for Describing a Motion

# Differentiation and Integration for Describing a Motion

Calculus is divided into two major concepts: differentiation and integration. Differentiation is a technique for analyzing small changes in one quantity concerning a unit change in another. Integration, on the other hand, is used to combine little and discrete data that cannot be combined separately and represented in a single value. The rate of change of speed concerning time is a real-life illustration of differentiation (i.e., velocity).

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In Physics we only need to know basic differentiation and integration for the derivation of the equation of motion as well as to solve numerically.

An equation of motion is a mathematical formula that represents a body’s position, velocity, or acceleration against a specific frame of reference. Newton’s second law of motion is the basic equation for classical mechanics, which states that the force F acting on a body is equal to the mass m of that body multiplied by the acceleration of that mass center,

F = ma. If the force acting on a body is known to be a function of time, Newton’s equation can potentially be used to deduce the velocity and position of the body as functions of time. A falling body, for example, accelerates at a constant rate of g. We have described more about the equations of motion in the latter part of this article.

## Topics Discussed

1. Definition of the equation of motion
2. Distance and Displacement
3. Derivation of Equation of motion

## Importance of equation of motion and use of differentiation and integration from JEE & NEET point of view

Distance: The term “distance” refers to the actual measurement of an object’s entire change in position over time. Because it is a scalar quantity, it just gives the magnitude.

Displacement: Displacement is the shortest measure of an object’s net change in position (during a certain time interval). Because it is a vector quantity, it has both a magnitude and a direction.

The displacement of an object is related to its velocity, acceleration, and time using equations of motion. An object’s motion can take many different pathways. Our focus here will be moved in one dimension (straight-line motion). As a result, we can only compute displacement, velocity, and acceleration magnitudes in the positive and negative directions, with negative values pointing in the opposite direction.

Kinematics equations of motion define the most fundamental ideas of object motion. The motion of an object in 1D, 2D, and 3D is governed by these equations. They make it simple to calculate expressions like an object’s position, velocity, or acceleration over time.

If there is no acceleration, the formula is as follows:

v=u+at

v = u + at is the first equation of motion. Here, v represents the final velocity, u represents the initial velocity, a represents the acceleration, and t represents the time. Velocity-time relations give the first equation of motion, which can be applied to find acceleration.

s=ut+12at

v2=u2+2as2

where, v= Velocity, u= initial velocity, a= acceleration, s=Displacement and t=time

The above three mentioned equations are known as the equation of motion.

## Derivation of First equation of motion

Objects moving in one dimension with constant acceleration have the following acceleration:

a=vt, where Δv is the change in the velocity of the object over a time Δt.

If an entity starts with a velocity of u at time t=0 and has a velocity of v at time t, then its velocity change is: v=v-u and t=t

Now substitute these values in the equation of acceleration

a=v-ut at=v-u

v=u+at

## Derivation of Second equation of method

Velocity= Displacement Time v=st

s=v.t

If the velocity is not constant, we may substitute average velocity for velocity in the previous equation and rewrite it as follows:

s=Initial velocity+Final Velocity2(t)

s=u+v2t

s=[u+(u+at)]2t

s=2u2+at2t

s=u+at22t

s=ut+12at2

## Derivation of third equation of motion

We know that v=u+at

t=v-ua

When we plug this into the equation for the change in the object’s displacement, we get:

s=ut+12at2

s=u(v-ua)+12a(v-ua)2

s=122uv-2u2+v2+u2-2uva

12v2u2a

that can be written as

v2=u2+2as

### Solved Examples

Question 1: Define Average Speed and Average Velocity?

Answer 1: Average Speed

It is defined as the total path covered in total time.

Avg. Speed = D/T

Average Velocity

The displacement times the time is the Total displacement.

Avg. Velocity= S2– S1/ T2– T1 = S/T

Question 2: A resting body was accelerated and traveled for 5 minutes. A constant acceleration of 2 meters per second was applied. Calculate the final speed at which the object came to a stop. Also, determine the final velocity using the equation of motion.

Answer 2:Finding out the final velocity can be determined with the help of the first equation of motion.

First equation of motion, v=u +at

Initial velocity= 0 m/sec

A 5 minute period of motion equals 5 seconds multiplied by 60 seconds equals 300 seconds

The constant acceleration provided to the object= 2 m/sec2

v= u+ at

After putting value the equation will be;

v= 0+ 2× 300

v= 600m/sec

## Importance of this topic from JEE & NEET point of view

Equation of motion, Differentiation, and Integration are the subjects that pique people’s curiosity in physics. Students may find it challenging to grasp the concepts at first, but once they do, learning and solving problems on these topics will be a lot of fun.

It is also critical from the standpoint of JEE & NEET because it creates the foundation for all mechanics.

The JEE Main Physics syllabus contains chapters such as Unit and measurement, thermodynamics, rotational motion, kinematics, properties of solids and liquids, work, energy and power, laws of motion, gravitation, electronic devices, oscillations and waves, current electricity, kinetic theory of gasses, electromagnetic induction and alternating currents, communication systems, magnetic effects of current and magnetism, electromagnetic waves, optics, atoms and nuclei, dual nature of matter, and radiation, and electrostatics.

You can expect 1-2 questions in JEE main from the Kinematics topic.

The physics section is termed as a daunting section of NEET exams over the years by candidates as it involves conceptual clarity and a solid understanding of physics concepts to apply the acquired knowledge on numerical-based questions. This section is often termed tricky and long as it involves calculations and reasoning.

You can expect 4-5 questions from topics like the equation of motion that comes under the Kinematics chapter.

Also read: NEET Exam Pattern 2022

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