Question

A particle starts from rest from origin and is moving in $XY$-plane, its component of acceleration in the direction of instantaneous velocity is $a_0=aâ‹\dots \hat{v}$. where $a$ is a constant vector of magnitude $2m{s}^{−2}$ directed along $Y$-axis. Find its speed in $m{s}^{−1}$ at $y=1m$.

Answer 1

Answer: 2

$a_0=âˆ\pounds aâˆ\yen vâˆ\pounds \cos θ$ or $a_0=a\cos θ$
Or $v\frac{dv}{ds}=a\cos θ$ or $vdv=ads\cos θ$
$âˆ\mu s=x\hat{i}+y\hat{j}$ or $ds=dx\hat{i}+dy\hat{j}$
$∴vdv=aâ‹\dots ds=a\hat{j}â‹\dots (dx\hat{i}+dy\hat{j})$
Or $\underset{0}{\overset{v}{∫}}vdv=ady=2\underset{0}{\overset{y}{∫}}dy\left(âˆ\mu a=2m{s}^{−2}\right)$
Or $\frac{v^2}{2}=2y$
Or $v=\sqrt{4y}=\sqrt{4×1}=2m{s}^{−1}$