Question
$$A$$ and $$B$$ are moving in 2 circular orbits with angular velocity $$2\omega $$ and $$\omega $$ respectively. Their positions are as shown at $$t = 0.$$ Find the time when they will meet for the first time.
$$A$$ and $$B$$ are moving in 2 circular orbits with angular velocity $$2\omega $$ and $$\omega $$ respectively. Their positions are as shown at $$t = 0.$$ Find the time when they will meet for the first time.
A.
$$\frac{\pi }{{2\omega }}$$
B.
$$\frac{{3\pi }}{{2\omega }}$$
C.
$$\frac{\pi }{\omega }$$
D.
they will never meet
Answer :
they will never meet
Solution :
Case 1:
When they rotate in same sense $$2m\pi = 2\omega t$$
$$\eqalign{ & \frac{{3\pi }}{2} + 2n\pi = \omega t;2m\pi = 2\left( {\frac{{3\pi }}{2} + 2n\pi } \right) \cr & 2m = 3 + 4n;m = \frac{3}{2} + 2n \Rightarrow m - 2n = \frac{3}{2} \cr} $$
Not possible for $$m$$ and $$n$$ being integer.
Case 2 :
When they rotate in opposite sense $$2m\pi = 2\omega t$$
$$\eqalign{ & \frac{\pi }{2} + 2n\pi = \omega t;2m\pi = 2\left( {\frac{\pi }{2} + 2n\pi } \right) \cr & 2m\pi = \pi + 4n\pi ;2m - 4n = 1 \cr} $$
Not possible for $$m$$ and $$n$$ integer.
Case 1:
When they rotate in same sense $$2m\pi = 2\omega t$$
$$\eqalign{ & \frac{{3\pi }}{2} + 2n\pi = \omega t;2m\pi = 2\left( {\frac{{3\pi }}{2} + 2n\pi } \right) \cr & 2m = 3 + 4n;m = \frac{3}{2} + 2n \Rightarrow m - 2n = \frac{3}{2} \cr} $$
Not possible for $$m$$ and $$n$$ being integer.
Case 2 :
When they rotate in opposite sense $$2m\pi = 2\omega t$$
$$\eqalign{ & \frac{\pi }{2} + 2n\pi = \omega t;2m\pi = 2\left( {\frac{\pi }{2} + 2n\pi } \right) \cr & 2m\pi = \pi + 4n\pi ;2m - 4n = 1 \cr} $$
Not possible for $$m$$ and $$n$$ integer.
