Question

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

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

Answer 1

The correct option is B $7.5R$

Let the sphere collide after time t.

when the smaller sphere covered the distance $x_1$ and bigger sphere covered distance $x_2$

The gravitational force acting between $2$ sphere depends on the distance which is a variable quantity.

The gravitational force $f\left(x\right)=\frac{Gm\times 5m}{(12R-x{)}^2}$

Acceleration of smaller body, $a_1\left(x\right)=\frac{G5m}{(12R-x{)}^2}$

Acceleration of Bigger Body, $a_2\left(x\right)=\frac{Gm}{(12R-x{)}^2}$

From the equation of motion

$x_1=\frac{1}{2}{a}_1\left(x\right)t^2$ and $x_2=\frac{1}{2}{a}_2\left(x\right)t^2$

$\Rightarrow \frac{x_1}{x_2}=\frac{a_1}{a_2}\frac{\left(x\right)}{\left(x\right)}=5\Rightarrow x_1=5x_2$

$x_1+x_2=9R$

$\Rightarrow x_1+\frac{x_1}{5}=9R$

$\Rightarrow x_1=\frac{45R}{6}=7.5R$

Smaller sphere covered a distance of $7.5R$ before collision.