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The position of particle at any instant is given by straight r with rightwards arrow on top equals 50 straight t straight j with hat on top plus 5 straight t squared straight i with hat on topAt what instant the velocity vector will make an angle 45° with X-axis?

Given,

Position of the particle, straight r with rightwards arrow on top space equals space 50 straight t straight j with hat on top plus 5 straight t squared straight i with hat on top

∴ Velocity, 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 left parenthesis 50 straight t space straight j with hat on top plus 5 straight t squared straight i with hat on top right parenthesis 

                 equals 50 straight j with hat on top plus 10 straight t space straight i with hat on top

The angle which the velocity vector makes with X-axis at any instant is,

tanθ space equals space straight v subscript straight y over straight v subscript straight x equals fraction numerator 50 over denominator 10 straight t end fraction 

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Therefore,

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rightwards double arrow          straight t equals 5 straight s

So, when time is 5 seconds, the velocity vector will make an angle 450 with x-axis. 

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