Question

Two spherical bodies of mass $M$ and $5M$ and radii $R$ and $2R$, 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 the collision is:

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

Answer 1

The correct option is C

$7.5R$

Step 1: Given

Mass of first sphere: $m_1=M$

Mass of second sphere: $m_2=5M$

Radius of first sphere: $r_1=R$

Radius of second sphere: $r_2=2R$

Initial distance between them: $d=12R$

Step 2: Formula Used

$m_1{d}_1=m_2{d}_2$

Step 3: Find the distance covered by both spheres

Calculate the distance between the centers spheres when they collide by adding the radii of both, as when they collide they will be in contact and distance between their centers will be equal to sum of their radii.

$\begin{array}{rcl}Distancebetweencenters& =& R+2R\\ & =& 3R\end{array}$

Calculate the distance travelled by the spheres by subtracting the distance at time of collision from the initial distance.

$\begin{array}{rcl}Distancetravelled& =& 12R-3R\\ & =& 9R\end{array}$

Step 4: Find the distance covered by smaller sphere

Assume the distance covered by the smaller sphere to be $x$. This implies that the distance covered by bigger sphere is $9R-x$. Calculate the value of $x$ by using the formula.

$$\begin{gathered} m_1{d}_1=m_2{d}_2 \\ \Rightarrow Mx=5M\left(9R-x\right) \\ \Rightarrow x=45R-5x \\ \Rightarrow 6x=45R \\ \Rightarrow x=7.5R \end{gathered}$$

Hence, the distance covered by smaller body is $7.5R$.