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Motion in A Plane

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Physics Part I

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Physics

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Class 10 Class 12

A particle moves in XY plane, such that its position at any instant is
      straight x equals straight alpha space sin space ωt           and       straight y equals straight alpha left parenthesis 1 minus cos space ωt right parenthesis
Find the angle between velocity vector and acceleration vector at any instant.


The position of the particle at any instant is,

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Therefore the velocity of particle at any instant is,

      straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space straight d over dt open parentheses αsinωt straight i with hat on top plus straight alpha left parenthesis 1 minus cosωt right parenthesis straight j with hat on top close parentheses

rightwards double arrow  space straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space αω space cosωt straight i with hat on top plus αω space sinωt straight j with hat on top

Acceleration of particle at any instant is,

      straight a with rightwards arrow on top equals straight d over dt left parenthesis straight v with rightwards arrow on top right parenthesis equals straight d over dt open parentheses aω space cosωt space straight i with hat on top plus space αωsinωt space straight j with hat on top close parentheses

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Now,  

straight a with rightwards arrow on top times straight v with rightwards arrow on top equals left parenthesis negative αω squared sinωt straight i with hat on top plus αω squared cosωt straight j with hat on top right parenthesis times left parenthesis αω space cosωt space straight i with hat on top plus αωsinωt space straight j with hat on top right parenthesis
       equals negative straight alpha squared straight omega cubed sinωtcosωt plus straight alpha squared straight omega cubed cosωtsinωt equals 0

Since dot product of acceleration vector and velocity vector is zero, therefore the acceleration vector and velocity vector is 90 degree. 

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