A force \[\vec{F} = \vec{v} \times \vec{A}\] is exerted on a particle in addition to the force of gravity, where \[\vec{v}\] is the velocity of the particle and \[\vec{A}\] 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 without being defelected and with a constant velocity?

#### Solution

For the particle to move without being deflected and with constant velocity, the net force on the particle should be zero.

\[\vec{F} + m \vec{g} = 0\]

\[\Rightarrow \left( \vec{v} \times \vec{A} \right) + \vec{mg} = 0\]

\[ \Rightarrow \left( \vec{v} \times \vec{A} \right) = - \vec{mg}\]

\[\left| vA\sin\theta \right| = \left| mg \right|\]

\[\therefore v = \frac{mg}{A\sin\theta}\]

v will be minimum when sinθ = 1.

⇒ θ = 90°

\[\therefore v_{\text{min}} = \frac{mg}{A}\]