# SoLn.

Our Problem can be classified as an indirect problem of kinematics. Thus from (1.3) & (1.2) we have to move towards (1.1) .It is of utmost importance in any kind of problem to know your direction (even if you don't know the path.)

Using (1.3)

$\cfrac { dv }{ dt } =a \\ dv=adt \\ \int _{ u }^{ v }{ dv } =\int _{ 0 }^{ t }{ adt }$

As the time changes from 0 to t the velocity changes from u to v.So on the left hand side the summation is made over v from u to v whereas on the right hand side the summation is made on time from 0 to t. Evaluating the integrals we get

${ [v] }_{ u }^{ v }=a{ [t] }_{ 0 }^{ t } \\ v-u=at \\ v=u+at$

Using (1.2) the last equation may be written as

$\cfrac { dx }{ dt } =v=u+at \\ dx=(u+at)dt \\ \int _{ 0 }^{ x }{ dx } =\int _{ 0 }^{ t }{ (u+at)dt }$

At t=0 the particle is at x=0. As time changes from 0 to t the position changes from 0 to x.So on the left hand side the summation is made on position from 0 to x whereas on the right hand side the summation is made on time from 0 to t.Evaluating the integrals we get,

${ [x] }_{ 0 }^{ x }=\int _{ 0 }^{ t }{ udt } \int _{ 0 }^{ t }{ atdt } \\ x=u{ [t] }_{ 0 }^{ t }+a{ { [t }^{ 2 }/2] }_{ 0 }^{ t } \\ x=ut+(a{ t }^{ 2 })/2$

Using the above two derived expressions,

${ v }^{ 2 }={ (u+at) }^{ 2 } \\ ={ u }^{ 2 }+2uat+{ a }^{ 2 }{ t }^{ 2 } \\ ={ u }^{ 2 }+2a[ut+\cfrac { 1 }{ 2 } a{ t }^{ 2 }] \\ ={ u }^{ 2 }+2ax$

The equations

$v=u+at$

$x=ut+\frac { 1 }{ 2 } a{ t }^{ 2 }$

${ v }^{ 2 }={ u }^{ 2 }+2ax$

are used very frequently while solving problems in kinematics involving constant acceleration as one of the physical quantities.If,however,the acceleration isn't constant,these three equations are not useful.We, then, need other tools & procedures.

Note by Soumo Mukherjee
3 years, 7 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$