Derivations of equations of motion

Asked by Jaypan14 | 7th Jul, 2018, 06:08: PM

Expert Answer:

Equation of motion 1.  :-     v = u+a×t
 
where u and v are initial and final velocity respectively. acceleration is a and t is time taken by the object to change velocity from u to v.
 
Acceleration is defined as rate of change of velocity.
 
hence,    a = (v-u)/t  or v = u+a×t
 
Equation of motion 2,    S = u×t + (1/2)×a×t2
 
where S is the ditance travelled in t seconds with initial velocity u and acceleration a
 
Distance travelled  S = average velocity × time = begin mathsize 12px style fraction numerator open parentheses v plus u close parentheses over denominator 2 end fraction cross times t space................ left parenthesis 1 right parenthesis end style
let us substitute, v = u+a×t in eqn.(1),  then we have,   S = begin mathsize 12px style fraction numerator open parentheses u plus a cross times t space plus u close parentheses over denominator 2 end fraction cross times t space equals space u cross times t space plus space 1 half cross times a cross times t squared end style
Equation. 3     v2 = u2 + 2×a×S
 
eqn.(1) is rewritten here
Distance travelled  S = average velocity × time = begin mathsize 12px style fraction numerator open parentheses v plus u close parentheses over denominator 2 end fraction cross times t space................ left parenthesis 1 right parenthesis end style
let us substitute t using equation of motion (1),  t =(v-u)/a in the above eqn.(1)
 
the we have, begin mathsize 12px style S space equals space fraction numerator left parenthesis v plus u right parenthesis over denominator 2 end fraction cross times fraction numerator open parentheses v minus u close parentheses over denominator a end fraction space equals space fraction numerator open parentheses v squared space minus space u squared close parentheses over denominator 2 a end fraction space o r space space space v squared space equals space u squared space plus space 2 cross times a cross times S end style

Answered by Thiyagarajan K | 7th Jul, 2018, 06:31: PM

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