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Motion In Straight Line

Question
CBSEENPH11017219

The acceleration of a particle at any instant is given by a = X + Yv, where X and Y arc constants and v is velocity. Find the velocity of the particle at any instant. Assume that the particle starts from rest.

Solution

Given, the particle starts from rest. 

Acceleration, a = space space space fraction numerator dv over denominator space dt end fraction space equals space straight X space plus space Yv space 
rightwards double arrow space space space space space space space space space space space space space space space fraction numerator dv over denominator straight D space plus space Yu end fraction space equals space dt space



At time = 0; velocity = 0 and let v be the velocity at any instant t.
Therefore,

space space space space integral subscript 0 superscript straight v fraction numerator dv over denominator straight X space plus space Yv end fraction space equals space integral subscript 0 superscript straight t dt space

rightwards double arrow space 1 over straight Y integral subscript 0 superscript straight v fraction numerator straight Y. straight d over denominator straight X space plus space Yv end fraction space equals space integral subscript 0 superscript straight t dt space

rightwards double arrow 1 over straight Y space Ln space left parenthesis straight X space plus space Yv right parenthesis vertical line subscript 0 superscript straight v space equals space straight t vertical line subscript 0 superscript straight t space

rightwards double arrow space Ln space left parenthesis straight X plus Yv right parenthesis space minus space Ln left parenthesis straight X right parenthesis space equals space straight Y left parenthesis straight t minus 0 right parenthesis space

rightwards double arrow space Ln space open parentheses fraction numerator straight X plus Yv over denominator straight X end fraction close parentheses space equals space Yt space

rightwards double arrow space 1 space plus space Yv over straight X space equals space straight e to the power of straight n

rightwards double arrow space space straight v space equals space straight X over straight Y left parenthesis straight e to the power of straight n space minus space 1 right parenthesis