Particles A and B are moving with constant ve | vedclass

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(D) Let the initial positions of particles $A$ and $B$ be $(-x_0, 0)$ and $(0, -y_0)$ respectively,where $x_0, y_0 > 0$.Let their constant velocities

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(D) Let the initial positions of particles $A$ and $B$ be $(-x_0, 0)$ and $(0, -y_0)$ respectively,where $x_0, y_0 > 0$.Let their constant velocities be $\vec{v}_A = v_A \hat{i}$ and $\vec{v}_B = v_B \hat{j}$.At any time $t$,the positions are $\vec{r}_A(t) = (-x_0 + v_A t) \hat{i}$ and $\vec{r}_B(t) = (-y_0 + v_B t) \hat{j}$.The separation $S$ between them is given by $S = |\vec{r}_B - \vec{r}_A| = \sqrt{(v_B t - y_0)^2 + (x_0 - v_A t)^2}$.Squaring both sides,$S^2 = (v_B^2 + v_A^2) t^2 - 2(v_B y_0 + v_A x_0) t + (y_0^2 + x_0^2)$.This is a quadratic equation of the form $S^2 = at^2 + bt + c$,which represents a hyperbola. The graph of $S$ versus $t$ is a hyperbola that decreases to a minimum value and then increases,which corresponds to the curve shown in option $D$.

Particles $A$ and $B$ are moving with constant velocities along the $x$ and $y$ axes respectively. The graph of the separation $S$ between them with time $t$ is:

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