Question

Two uniform solid spheres of equal radii R, but mass M and 4M have a centre to centre separation 6 R, the two spheres are held fixed on a horizontal floor. A projectile of mass m is projected from the surface of the sphere of mass M directly towards the centre of the second sphere. Obtain an expression for the minimum speed v of the projectile so that it reaches the surface of the second sphere

• A. $v={\left(\frac{5GM}{3R}\right)}^{\frac{1}{2}}$
• B. $v={\left(\frac{3GM}{5R}\right)}^{\frac{1}{2}}$
• C. $v={\left(\frac{3R}{5GM}\right)}^{\frac{1}{2}}$
• D. $v={\left(\frac{8R}{5GM}\right)}^{\frac{1}{2}}$

Answer 1

Answer: (B)

$v={\left(\frac{3GM}{5R}\right)}^{\frac{1}{2}}$

From conservation of energy:
$\frac{1}{2}{v}^2-\frac{GM}{R}-\frac{4GM}{5R}=-\frac{GM}{2R}-\frac{GM}{R}$
$\frac{1}{2}{v}^2=\frac{3GM}{10R}$
$v={\left(\frac{3GM}{5R}\right)}^{\frac{1}{2}}$