# A Force → F = → V × → a → V is the Velocity of the Particle and → a → a is a Constant Vector in the Horizontal Direction. with What Minimum Speed, a Particle of Mass M Be Projected - Physics

Sum

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°
$\therefore v_{\text{min}} = \frac{mg}{A}$

Concept: Newton’s Second Law of Motion
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#### APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 5 Newton's Laws of Motion
Q 21 | Page 80