Question

A particle starts from rest from origin and is moving in $X Y$-plane, its component of acceleration in the direction of instantaneous velocity is $a_{0}=a \cdot \hat{v}$. where $a$ is a constant vector of magnitude $2\, ms ^{-2}$ directed along $Y$-axis. Find its speed in $ms ^{-1}$ at $y=1 \,m$.

Answer 1

$a_{0}=|a \| v| \cos \theta$ or $a_{0}=a \cos \theta$
Or $v \frac{d v}{d s}=a \cos \theta $ or $v d v=a d s \cos \theta$
$\because s=x \hat{i}+y \hat{j} $ or $d s=d x \hat{i}+d y \hat{j}$
$\therefore v d v=a \cdot d s=a \hat{j} \cdot(d x \hat{i}+d y \hat{j})$
Or $ \int\limits_{0}^{v} v d v=a d y=2 \int\limits_{0}^{y} d y \left(\because a=2\, ms ^{-2}\right)$
Or $ \frac{v^{2}}{2}=2 y$
Or $ v=\sqrt{4 y}=\sqrt{4 \times 1}=2\, ms ^{-1}$