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

Options

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

Answer

A

Explanation

Let the spheres collide at time t, after the smaller sphere covering a distance of x and the bigger sphere covering a distance of `x_2`.

The gravitational force acting between two spheres depends on the distance x which is a variable quantity.
The gravitational force, `F(x)= (GM xx 5M)/((12 R - x)^2)`
Acceleration of smaller body, ` a_1(x)= (G xx 5M)/((12R -x)^2)`
Acceleration of bigger body, `a_2(x) = (GM)/((12 R -x)^2)`
From equation of motion, `x_1 = 1/2 a_1(x)t^2 " and " x_2 =1/2 a_2(x) t^2`
` therefore x_1/x_2 =(a_1(x))/(a_2(x)) = 5 " or " x_1 = 5x_2`
As `x_1+x_2 =9R therefore x_1 + (x_1)/(5) = 9R " or " x_1 = (45R)/(6) = 7.5 R`
Therefore the two spheres collide when the smaller sphere covers a distance of 7.5 R.