ICSE Class 9 Answered
Derivations of equations of motion
Asked by Jaypan14 | 07 Jul, 2018, 18:08: PM
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](https://images.topperlearning.com/topper/tinymce/cache/fc24981f9ebef72a8e61f8bc4905bfeb.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/fc24981f9ebef72a8e61f8bc4905bfeb.png)
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](https://images.topperlearning.com/topper/tinymce/cache/86b2c281f895135dc0e1e178b2109808.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/86b2c281f895135dc0e1e178b2109808.png)
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](https://images.topperlearning.com/topper/tinymce/cache/fc24981f9ebef72a8e61f8bc4905bfeb.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/fc24981f9ebef72a8e61f8bc4905bfeb.png)
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](https://images.topperlearning.com/topper/tinymce/cache/598800b5c45a77ed23d22e8404c7ba00.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/598800b5c45a77ed23d22e8404c7ba00.png)
Answered by Thiyagarajan K | 07 Jul, 2018, 18:31: PM
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