# Motion

## 6. Equations of uniformly accelerated motion

• There are three equations of bodies moving with uniform acceleration which we can use to solve problems of motion

### First Equation of motion

• The first equation of motion is $v = u + at$ , where v is the final velocity and u is the initial velocity of the body.
• First equation of motion gives velocity acquired by body at any time $t$.
• Now we know that acceleration

so, $a = \frac{{v - u}}{t}$
and, $at = v - u$
rearranging above equation we get first equation of motion that is
$v = u + at$

### Second Equation of motion

• Second equation of motion is
$s = ut + \frac{1}{2}a{t^2}$
where $u$ is initial velocity, $a$ is uniform acceleration and $s$ is the distance travelled by body in time $t$.
• Second equation of motion gives distance travelled by a moving body in time $t$.
• To obtain second equation of motion consider a body with initial velocity $u$ moving with acceleration a for time $t$ its final velocity at this time be $v$. If body covered distance $s$ in this time $t$ , then average velocity of the body would be

Distance travelled by the body is

From first equation of motion
$v = u + at$
So putting first equation of motion in above equation we get ,
$s = \frac{{u + u + at}}{2} \times t = \frac{{\left( {2u + at} \right)t}}{2} = \frac{{2ut + a{t^2}}}{2}$
Rearranging it we get
$s = ut + \frac{1}{2}a{t^2}$

### Third equation of motion

• Third equation of motion is
${v^2} = {u^2} + 2as$ where $u$ is initial velocity, $v$ is the final velocity, $a$ is uniform acceleration and $s$ is the distance travelled by the body.
• This equation gives the velocity acquired by the body in travelling a distance $s$.
• Third equation of motion can be obtained by eliminating time t between first and second equations of motion.
So, first and second equations of motion respectively are
$v = u + at$ and $s = ut + \frac{1}{2}a{t^2}$
Rearranging first equation of motion to find time t we get
$t = \frac{{v - u}}{a}$
Putting this value of t in second equation of motion we get
$s = \frac{{u\left( {v - u} \right)}}{a} + \frac{1}{2}a{\left( {\frac{{v - u}}{a}} \right)^2}$

$s = \frac{{uv - {u^2}}}{a} + \frac{{a\left( {{v^2} + {u^2} - 2uv} \right)}}{{2{a^2}}}$

$s = \frac{{2uv - 2{u^2} + {v^2} + {u^2} - 2uv}}{{2a}}$

Rearranging it we get
${v^2} = {u^2} + 2as$
• These three equations of motion are used to solve uniformly accelerated motion problems and following three important points should be remembered while solving problems
1. if a body starts moving from rest its initial velocity $u = 0$
2. if a body comes to rest i.e., it stops then its final velocity would be $v = 0$
3. If a body moves with uniform velocity then its acceleration would be zero.

1)The __________________ equation of motion is v=u+at , where v is the final velocity and u is the initial velocity of the body.
2)Second equation of motion is  s=ut+1/2at2  where u is initial velocity, a is uniform acceleration and s is the ____________
______ travelled by body in time t.
3) Third equation of motion is v2=u2+2as where u is initial velocity, v is the final velocity, a is __________ and s is the distance travelled by the body.

Reference Books for class 9 science

Given below are the links of some of the reference books for class 9 science.

These books by S.Chand Publications are detailed in their content and are must have books for class 9 students. Some other books that might be worth considering are
1. Oswaal CBSE Question Bank Class 9 Science Chapterwise & Topicwise We all know importance of practicing what we have studied. This book is perfect for practicing what you have learned and studied in the subject.
2. Pearson Foundation Series (IIT-JEE/NEET) Physics, Chemistry, Maths & Biology for Class 9 (Main Books) | PCMB Combo These Foundation books would be helpful for students who want to prepare for JEE/NEET exams. Only buy them if you are up for some challenge and have time to study extra topics. These might be tough for you and you might need extra help in studying these books.
3. Foundation Science Physics for Class - 9 by H.C. Verma This is one of my favorite Physics book for class 9. Most of the book is within the limits of CBSE syllabus. It might overwhelm you with its language but I feel if you can understand the content is authentic with plenty of problems to solve.

You can use above books for extra knowledge and practicing different questions.

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