Question

A force $\overrightarrow{F}=\overrightarrow{v}\times \overrightarrow{A}$ is exerted on a particle in addition to the force of gravity, where $\overrightarrow{v}$ is the velocity of the particle and $\overrightarrow{A}$ is a constant vector in the horizontal direction. With what minimum speed a particle of mass $m$ be projected so that it continues to move undeflected with a constant velocity?

Answer 1

Given that, $\overrightarrow{F}=\overrightarrow{u}\times \overrightarrow{A}$ and $\overrightarrow{mg}$ act on the particle

For the particle to move undeflected with constant velocity, net force should be zero.

$\therefore (\overrightarrow{u}\times \overrightarrow{A})+\overrightarrow{mg}=0$

$\therefore (\overrightarrow{u}\times \overrightarrow{A})=\overrightarrow{-mg}$

Because, $(\overrightarrow{u}\times \overrightarrow{A})$ is perpendicular to the plane containing $\overrightarrow{u}$ and $\overrightarrow{A},\overrightarrow{u}$ should be the $xz-$ plane

Again $uA\sin \theta =mg$

$u=\frac{mg}{A\sin \theta }$

$u$ will be minimum when $\sin \theta =1\Rightarrow \theta ={90}^o$

$\therefore U_{min}=\frac{mg}{A}$ along $Z-$ axis.