Question

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

Answer 1

Given,

$F=v\times A$

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

$v\times A+mg=0$

$\therefore v\times A=-mg$

Because $v\times A$ is perpendicular to the plane containing $v$ and $A$, $v$ should be in the xz-plane.

Again, $vA\sin \theta =mg$

$\therefore v=\frac{mg}{A\sin \theta }$

$v$ will be minimum, when $\sin \theta =1\Rightarrow \theta ={90}^{\circ }$

$\therefore v_{min}=\frac{mg}{A}$ along z-axis