Two particles are simultaneously projected in the horizontal dire… - Vidyadip

MCQ

Two particles are simultaneously projected in the horizontal direction from a point Pat a certain height. The initial velocities of the particles are oppositely directed to each other and have magnitude v each. The separation between the particles at a time when their position vectors (drawn from the point P) are mutually perpendicular, is

A
$\frac{v^2}{2 g}$
B
$\frac{v^2}{g}$
✓
$\frac{4 v^2}{g}$
D
$\frac{2 v^2}{g}$

✓

Answer

Correct option: C.

$\frac{4 v^2}{g}$

(C)
$\overrightarrow{ r }_1= vt \hat{ i }-\frac{1}{2} gt ^2 \hat{ j }, \overrightarrow{ r }_2= vt (-\hat{ i })-\frac{1}{2} gt ^2 \hat{ j }$
Since both are perpendicular,
$\overrightarrow{ r }_1 \cdot \overrightarrow{ r }_2=0$
$\Rightarrow- v ^2 t ^2+\frac{1}{4} g^2 t ^4=0$
$\therefore \quad v ^2=\frac{1}{4} g^2 t ^2$
$\therefore \quad v =\frac{ gt }{2}$
$\therefore \quad t =\frac{2 v }{ g }$
$\therefore \quad \Delta x =2 vt =2 \times v \times \frac{2 v }{ g }=\frac{4 v ^2}{g}$

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