Equations of Motion by Graphical Method | Motion | Notes | Summary - Zigya

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Motion

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Equations of Motion by Graphical Method

When an object moves along a straight line with uniform acceleration, it is possible to relate its velocity, acceleration during motion and the distance covered by it in a certain time interval by a set of equations known as the equations of motion. There are three such equations.

These are:

v = u + at

straight s space equals space ut space plus space 1 half space at squared

2 as space equals space straight v squared space minus straight u squared

Equation For Position

Graphical Derivation :

Suppose a body has initial velocity ‘u’ (i.e., the velocity at time t = 0 sec.) at

point ‘A’ and this velocity changes to ‘v’ at point ‘B’ in ‘t’ secs. i.e., final velocity will be ‘v’.

For such a body there will be an acceleration

straight a space equals space fraction numerator change space in space velocity over denominator change space in space time end fraction
straight a space equals space fraction numerator OB minus OA over denominator OC minus 0 end fraction space equals fraction numerator straight v minus straight u over denominator straight t minus 0 end fraction
space or space straight a space equals space fraction numerator straight v minus straight u over denominator straight t end fraction
or space straight v space equals space straight u space plus at

second Equation

straight s space equals ut space plus space 1 half space at squared

Distance travelled by object = Area of OABC (trapezium)

= Area of OADC (rectangle) + Area of ΔABD

OA space straight x space AD space plus space 1 half space straight x space AD space straight x space BD
equals space straight u space straight x space straight t space plus 1 half space straight x space straight t space straight x space left parenthesis straight v minus straight u right parenthesis
equals space ut space plus space 1 half space straight x space straight t space space straight x space at
open parentheses because space fraction numerator straight v minus straight u over denominator straight t end fraction space equals straight a close parentheses space
straight s space equals space ut space plus space 1 half at squared

Third equation

straight v squared space equals space straight u squared space plus space 2 as
straight s space equals space Area space of space trapezium space OABC
straight s space equals space fraction numerator left parenthesis OA space plus space BC right parenthesis space straight x space OC over denominator 2 end fraction
straight s space equals space fraction numerator left parenthesis straight u space plus straight v right parenthesis space straight x space straight t over denominator 2 end fraction
straight s space equals space open parentheses fraction numerator straight u space plus straight v over denominator 2 end fraction close parentheses space straight x space open parentheses fraction numerator straight v minus straight u over denominator straight a end fraction close parentheses
or
open parentheses because space fraction numerator straight v minus space straight u over denominator straight t end fraction space equals straight a close parentheses

straight s space equals space fraction numerator straight v squared space minus straight u squared over denominator 2 straight a end fraction

or space straight v squared space equals space straight u squared space plus space 2 as

Equation For Position-Time Relation

position time relation

Equation For Velocity-Time Relation

Velocity -Time relation

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