- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
With the help of graph, derive the relation between:
a. velocity time
b. position time
c. position velocity
(derive three equation of motion)
Velocity-Time
A car having initial velocity u is being subjected to a uniform acceleration a for a time period of t, after the time t the final velocity of the car is v.
Now we need to find the relation between v,u,a,t graphically.
Initial velocity u at A = OA
Final velocity is V
Slope of velocity time graph is equal to acceleration a.
$a\ =\ \frac{v\ -\ u}{t\ -\ 0}$
$\Longrightarrow \ v\ =\ u\ +\ at$
Position -Time
Consider the linear motion of a body with an initial velocity u. Let the body accelerate uniformly and acquire a final velocity v after time t. The velocity–time graph is a straight line AB as shown below.
At t = 0, initial velocity = u = OA
At t = t, final velocity = v = OC
The distance S traveled in time t = area of the trapezium OABD
$s\ =\ \frac{1}{2} \ \times \ ( OA\ +\ DB) \ \times \ OD$
$s\ =\ \frac{1}{2} \ \times \ ( u\ +\ v) \ \times \ t$
Since v = u + at,
$s\ =\ \frac{1}{2} \ \times \ ( u+\ u\ +\ at) \ \times \ t$
$s\ =\ ut\ +\ \frac{1}{2} at^{2}$
Velocity- Position
From, velocity time relation
$t\ =\ \frac{v\ -\ u}{a}$
substituting the value of 't' in equation $s\ =\ \frac{1}{2} \ \times \ ( u\ +\ v) \ \times \ t$
we get,
$s\ =\ \left(\frac{v\ +\ u}{2}\right) \ \times \ \left(\frac{v\ -\ u}{2}\right)$
=> v²=u²+2as