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Newton’s Laws of Motion — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsNewton’s Laws of Motion3 Marks
Question
A force $\overrightarrow{\text{F}}=\overrightarrow{\text{v}}\times\overrightarrow{\text{A}}$ is exerted on a particle in addition to the force of gravity, where v is the velocity of the particle and $\overrightarrow{\text{A}}$ 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?
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Answer

Given that, $\overrightarrow{\text{F}}=\overrightarrow{\text{u}}\times\overrightarrow{\text{A}}$ and $\overrightarrow{\text{mg}}$ act on the particle. For the particle to move undeflected with constant velocity, net force should be zero.$\therefore\big(\overrightarrow{\text{u}}\times\overrightarrow{\text{A}}\big)+\overrightarrow{\text{mg}}=0$
$\therefore\big(\overrightarrow{\text{u}}\times\overrightarrow{\text{A}}\big)-\overrightarrow{\text{mg}}=0$
Because, $\big(\overrightarrow{\text{u}}\times\overrightarrow{\text{A}}\big)$ is perpendicular to the plane containing $\overrightarrow{\text{u}}$ and $\overrightarrow{\text{A}},\overrightarrow{\text{u}}$ should be in the xz-plane. Again, $\text{u A}\sin\theta=\text{mg}$$\therefore\text{u}=\frac{\text{mg}}{\text{A}\sin\theta}$
u will be minimum, when $\sin\theta=1\Rightarrow\theta=90^{\circ}$$\therefore\text{u}_{\text{min}}=\frac{\text{mg}}{\text{A}}$ along Z-axis.

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