Question

Two projectiles $A$ and $B$ are thrown simultaneously towards each other. If the initial distance between them is $30\text{m}$ and their initial velocities are $V_A=2\hat{i}-3\hat{j}$ and $V_B=5\hat{i}+\hat{j}$. Then the minimum distance between them during their flight will be.

• A. $25\text{m}$
• B. $5\text{m}$
• C. $24\text{m}$
• D. $17\text{m}$

Answer 1

The correct option is C $24\text{m}$

From ground frame,
${\overrightarrow{V}}_{AG}=2\hat{i}-3\hat{j}$
${\overrightarrow{V}}_{BG}=5\hat{i}+\hat{j}$
From ${}^{\prime }{A}^{\prime }$ frame:
${\overrightarrow{V}}_{BA}={\overrightarrow{V}}_{BG}-{\overrightarrow{V}}_{AG}$
$=5\hat{i}+\hat{j}-(2\hat{i}-3\hat{j})$

${\overrightarrow{V}}_{BA}=3\hat{i}+4\hat{j}$
$d_{\text{short}}=d\sin \theta$
$=30\sin \theta$
$\tan \theta =\frac{4}{3}$
$\Rightarrow \sin \theta =\frac{4}{5}$
$d_{\text{short}}=30\times \frac{4}{5}$
$=24\text{m}$