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