Two particles are projected from the same point , with the same speed, in the same vertical plane, at different angles with the horizontal. A frame of reference is fixed to one
particle. The position vector of the other particle as observed from this frame is $\overset{⇀}{r}$ Which of the follwing satements are correct?

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direction of $\overset{⇀}{r}$ does not change

$\overset{⇀}{r}$ changes in magnitude and direction with time

The magnitude of $\overset{⇀}{r}$ increases linearly with time

The direction of $\overset{⇀}{r}$ changes with time; its magnitude may or may not change, depending on the angles of projection

answer is A, C.

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Detailed Solution

Let u = the speed of projection, $θ_{1}$ and $θ_{2}$ = the angles of projection. Let $\overset{⇀}{r_{1}}$ and ${\overset{⇀}{r}}_{2}$ be the position vectors of the two particles in a ground frame.

$r_{1} = (ucos θ_{1}); t \hat{i} + [(u \sin θ_{1}) t - \frac{1}{2} g t^{2}] \hat{j}$

$\vec{r_{2}} = {(ucos θ_{2})}^{t} \hat{i} + [(u \sin θ_{2}) t - \frac{1}{2} g t^{2}] \hat{j}$

The position vector of one particle with respect to another is $\vec{r} = {\vec{r}}_{1} - \vec{r_{2}} = [u (\cos θ_{1} - \cos θ_{2}) t] \hat{i} + [u (\sin θ_{1} - \sin θ_{2}) t] \hat{j} = a t \hat{i} + b t \hat{j} = [a \hat{i} + b \hat{j}] t$ . wher $α$ and $β$ are constants.