Question

# Two masses $$M_1$$ and $$M_2$$ at an infinite distance from each other and initially at rest, start interacting gravitationally. Find their velocity of approach when they are distances apart.

Solution

## Since they move under mutual attraction and no external force acts on them, their momentum and energy are conserved. Therefore,$$\therefore 0=\dfrac{1}{2}M_1v^2_1+\dfrac{1}{2}M_2v^2_2-\dfrac{GM_1M_2}{s}$$It is zero because in the beginning, both kinetic energy and potential energy are zero.$$0=M_1v_1+M_2v_2$$Solving the equations,$$v^2_1=\dfrac{2GM^2_2}{s(M_1+M_2)}$$and $$v^2_2=\dfrac{2GM^2_1}{s(M_1+M_2)}$$V(velocity of approach)$$=v_1-(-v_2)=v_1+v_2$$$$=\sqrt{\dfrac{2G(M_1+M_2)}{s}}$$Physics

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