Question 243: Gravitation

Question

Two spherical bodies of mass $$M$$ and $$5M$$ & radii $$R$$ & $$2R$$ respectively are released in free space with initial separation between their centers equal to $$12R$$. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is

Accepted Answer

Let $$t$$ be the time taken for the two masses to collide and $${x_{5M,}}\,{x_M}$$ be the distance travelled by the mass $$5M$$ and $$M$$ respectively.
The gravitational force acting between two sphere when the distance between them (12R - x) where x is a variable,
$$F = {{GM 	imes 5M} \over {{{\left( {12R - x} ight)}^2}}}$$
Acceleration of mass M, $${a_M} = {{G 	imes 5M} \over {{{\left( {12R - x} ight)}^2}}}$$
Acceleration of mass 5M, $${a_{5M}} = {{GM} \over {{{\left( {12R - x} ight)}^2}}}$$
For mass $$5M$$
$$u = 0,\,\,S = {x_{5M}},\,\,t = t,\,\,a{ = a_{5M}}$$
$$S = ut + {1 \over 2}a{t^2}$$
$$	herefore$$ $${x_{5M}} = {1 \over 2}{a_{5M}}{t^2}\,\,\,\,\,\,\,\,\,\,...\left( {i} ight)$$
For mass $$M$$
$$u = 0,\,\,s = {x_M},\,\,t = t,\,\,a = {a_M}$$
$$	herefore$$ $$s = ut + {1 \over 2}a{t^2} \Rightarrow $$
$${x_M} = {1 \over 2}{a_M}{t^2}\,\,\,\,\,\,\,\,\,...\left( {ii} ight)$$
Dividing $$(ii)$$ by $$(iii)$$
$${{{x_{5M}}} \over {{x_M}}} = {{{1 \over 2}{a_5}_M{t^2}} \over {{1 \over 2}{a_M}{t^2}}}$$
$$ = {{{a_{5M}}} \over {{a_M}}} = {1 \over 5}$$
$$	herefore$$ $$5{x_{5M}} = {x_M}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left....( {iii} ight)$$
From the figure,
$${x_{5M}} + {x_M} = 12R - 2R - R = 9R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( iv ight)$$
From $$(iii)$$ and $$(iv)$$
$${{{x_M}} \over 5} + {x_M} = 9R$$
$$	herefore$$ $$6{x_M} = 45R$$
$$	herefore$$ $${x_M} = {{45} \over 6}R = 7.5R$$
So two sphere collide when the sphere of mass M covered the distance of 7.5R.

Answer

$$2.5$$ $$R$$

Answer

$$4.5$$ $$R$$

Answer

$$7.5$$ $$R$$

Answer

$$1.5$$ $$R$$

Gravitation

Two spherical bodies of mass $M and 5M & radii R & 2R respectively are released in free space with initial separation between their centers equal to 12R$. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is

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