Question
With what minimum speed should $$m$$ be projected from point $$C$$ in presence of two fixed masses $$M$$ each at $$A$$ and $$B$$ as shown in the figure such that mass $$m$$ should escape the gravitational attraction of $$A$$ and $$B$$
With what minimum speed should $$m$$ be projected from point $$C$$ in presence of two fixed masses $$M$$ each at $$A$$ and $$B$$ as shown in the figure such that mass $$m$$ should escape the gravitational attraction of $$A$$ and $$B$$
A.
$$\sqrt {\frac{{2GM}}{R}} $$
B.
$$\sqrt {\frac{{2\sqrt 2 GM}}{R}} $$
C.
$$2\sqrt {\frac{{GM}}{R}} $$
D.
$$2\sqrt 2 \sqrt {\frac{{GM}}{R}} $$
Answer :
$$\sqrt {\frac{{2\sqrt 2 GM}}{R}} $$
Solution :
$$\eqalign{ & \frac{1}{2}mv_{\min }^2 - \frac{{GMm}}{{R\sqrt 2 }} \times 2 = 0 \cr & \Rightarrow {v_{\min }} = \sqrt {\frac{{2GM\sqrt 2 }}{R}} \cr} $$
$$\eqalign{ & \frac{1}{2}mv_{\min }^2 - \frac{{GMm}}{{R\sqrt 2 }} \times 2 = 0 \cr & \Rightarrow {v_{\min }} = \sqrt {\frac{{2GM\sqrt 2 }}{R}} \cr} $$