Question

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

• A. $2.5R$
• B. $4.5R$
• C. $7.5R$
• D. $1.5R$

Answer 1

The correct option is C $7.5R$


Froms Newtons law of Gravitation


$F=\frac{Gm\left(5m\right)}{{(12R-x)}^2}=m{a}_{small}=5m{a}_{big}\Rightarrow a_{small}=\frac{G\left(5m\right)}{{(12R-x)}^2}\Rightarrow a_{big}=\frac{G\left(m\right)}{{(12R-x)}^2}$
From Equation of motion
$\therefore x=\frac{1}{2}{a}_{\text{small}}{t}^2=\frac{1}{2}\frac{G\left(5m\right)}{{(12R-x)}^2}{t}^2.....\left(1\right)$

$9R-x=\frac{1}{2}{a}_{\text{big}}{t}^2=\frac{1}{2}\frac{G\left(m\right)}{{(12R-x)}^2}{t}^2.....\left(2\right)$

$\left(1\right)÷\left(2\right)\Rightarrow x=7.5R$