Question

A particle moves in the x-y plane with the velocity $\overrightarrow{v}=a\hat{i}+bt\hat{j}$. AT the instant t=$a\sqrt{3}/b$ the magnitude of tangential, normal and total acceleration are _______, _________, & ________.

Answer 1

$\overrightarrow{a_{net}}=\frac{\overrightarrow{dv}}{dt}=b\hat{j}v=speed=\sqrt{a^2+b^2+t^2}{a}_t=\frac{dv}{dt}=\frac{1}{2\sqrt{a^2+b^2+t^2}}(b^2\times 2t)butt=\frac{a\sqrt{3}}{b}{a}_t=\frac{b^2\times \frac{a\sqrt{3}}{b}}{\sqrt{a^2+b^2\times \frac{a^2\times 3}{b^2}}}=\frac{b\sqrt{3}}{2}\overrightarrow{a_{net}}=\overrightarrow{a_t}+\overrightarrow{a_n}anddirectionof\overrightarrow{a_t}\&\overrightarrow{a_n}willbeperpendiculartoeachother\overrightarrow{a_{net}}=\sqrt{a_t^2+a_n^2}b=\sqrt{{\left(\frac{b\sqrt{3}}{2}\right)}^2+a_n^2}{a}_n^2=b^2-\frac{3b^2}{4}=\frac{b^2}{4}{a}_n=\frac{b}{2}$