EQUATIONS OF MOTION

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Let us consider an object moving with initial velocity (u), after time (t), it attains final velocity (v), covering distance (s) with uniform acceleration (a). 

 First equation of motion-

VELOCITY-TIME RELATION (v=u+at)


OA= u 
BC= v 
BD= v-u 
OC= t
                  Acceleration= Change in velocity ÷
                                            Time taken 
        a= BC-DC / OC = BD/OC 
        a= v-u/t (cross multiplication)  
    u+at =v (arrange it according to equation)

Hence, first equation v=u+at is proved. 

Second equation of motion 

POSITION-TIME RELATION (s=ut+½ at²)


 S= Area under the graph
S= Area of rect.(OADC)+Area of triangle ABD
               S=OA×OC+½ AD×BD 
               S= ut+½(t) (v-u)  
Put the values of (v-u) from eq 1 [v=u+at]
                                                            [v-u=at]
s=ut+½(t)(at) (arrange according to eq)

Hence, second equation s=ut+½at² is proved. 

Third equation of motion 

POSITION-VELOCITY RELATION (2as=v²-u²)


    S= Area under the graph 
    S= Area of trapezium OABDC
    S= ½ (Sum of all parallel sides) × Height 
    S= ½ (BC+OA) × OC
    S= ½ (v+u) (t)
Put the value of the from eq 1 [v=u+at]
                                                        [v-u=at]
                                                        [v-u/a=t]
S= ½(v+u)(v-u/a)
S= 1/2a (v+u) (v-u) [a+b][a-b]= a²-b²
2as =(v+u)(v-u) 

Hence, third equation 2as=v²-u² is proved. 



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