Physics - Gravitation Question with Solution | TestHub

Let the spheres collide after time$\mathit{\text{t}}$, when the smaller sphere covered distance${\mathit{\text{x}}}_1$and bigger sphere covered distance${\mathit{\text{x}}}_2$.

The gravitational force acting between two spheres depends on the
distance which is variable quantity.

The gravitational force,$\mathit{\text{F}}\left(\mathit{\text{x}}\right)=\frac{\mathit{\text{GM}}\times 5\mathit{\text{M}}}{{\left(\text{12}\mathit{\text{R}}-\mathit{\text{x}}\right)}^2}$

Accelration of smaller body,${\mathit{\text{a}}}_1\left(\mathit{\text{x}}\right)=\frac{\mathit{\text{G}}\times 5\mathit{\text{M}}}{{\left(\text{12}\mathit{\text{R}}-\mathit{\text{x}}\right)}^2}$

Acceleration of bigger body,${\mathit{\text{a}}}_2\left(\mathit{\text{x}}\right)=\frac{\mathit{\text{G}}\mathit{\text{M}}}{{\left(\text{12}\mathit{\text{R}}-\mathit{\text{x}}\right)}^2}$

From equation of motion,

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

$\Rightarrow \frac{{\mathit{\text{x}}}_1}{{\mathit{\text{x}}}_2}=\frac{{\mathit{\text{a}}}_1\left(\mathit{\text{x}}\right)}{{\mathit{\text{a}}}_2\left(\mathit{\text{x}}\right)}=5\Rightarrow {\mathit{\text{x}}}_1=5{\mathit{\text{x}}}_2$



We know that${\mathit{\text{x}}}_1+{\mathit{\text{x}}}_2=9\mathit{\text{R}}$

${\mathit{\text{x}}}_1+\frac{{\mathit{\text{x}}}_1}{5}=9\mathit{\text{R}}$$\therefore$${\mathit{\text{x}}}_1+\frac{\text{45}\mathit{\text{R}}}{5}=7.5\mathit{\text{R}}$

Therefore, the two spheres collide when the smaller sphere covers a distance of$\text{7.5}\mathit{\text{R}}$.