Question

Two particles of masses m & M are initially at rest at an infinite distance apart. They move towards each other and gain speeds due to gravitational attraction. Find their speeds when separation between the masses becomes equal to d.

Answer 1

To find the speeds of two particles of masses m and M when the separation between them becomes equal to d, we can use the principle of conservation of energy and solve for v_m and v_M using the equation 0.5mv_m^2 + 0.5MV_M^2 = GMm/d. The speeds are given by v_m = √(2GM/d)*(M/(M+m)) and v_M = √(2GM/d)*(m/(M+m)).

Step-by-step explanation:

In this problem, we have two particles of masses m and M initially at rest and then moving towards each other due to gravitational attraction. We need to find their speeds when the separation between them becomes equal to d.

Let's denote vm as the speed of the particle with mass m and vM as the speed of the particle with mass M.

Using the principle of conservation of energy, we can equate the initial potential energy of the system to the final kinetic energy. The initial potential energy is zero because the particles are initially at an infinite distance apart. The final kinetic energy is given by:

0.5mvm2 + 0.5MVM2 = GMm/d

Simplifying the equation and solving for vm and vM, we get:

vm = √(2GM/d)*(M/(M+m))

vM = √(2GM/d)*(m/(M+m))