Problem:

Two spheres of masses M and 5M, with radii R and 2R respectively, are allowed to move under the force of gravitation. The initial separation between their centers is 12R. Calculate the distance traveled by the smaller mass before collision.

Solution:

To solve this problem, we need to consider the gravitational force between the two spheres and determine the path of each sphere until they collide.

Step 1: Calculating Gravitational Force

The gravitational force between two objects can be calculated using Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

In this case, the mass of the smaller sphere is M and the mass of the larger sphere is 5M. The distance between their centers is initially 12R.

Step 2: Determining the Path of Each Sphere

Since the gravitational force is an attractive force, the smaller sphere will experience a force towards the larger sphere. As a result, the smaller sphere will move towards the larger sphere.

To determine the path of the smaller sphere, we need to consider the acceleration it experiences due to the gravitational force. The acceleration can be calculated using Newton's second law:

F = m * a

where F is the force, m is the mass, and a is the acceleration.

In this case, the force acting on the smaller sphere is the gravitational force, and its mass is M. Therefore, the acceleration of the smaller sphere is given by:

a = F / m = G * (m1 * m2) / (m * r^2)

Step 3: Calculating the Distance Traveled

To calculate the distance traveled by the smaller sphere before collision, we need to integrate the acceleration with respect to time. However, since we don't have the time, we can use a different approach.

Let's assume that the smaller sphere travels a distance x before collision. At that point, the distance between the centers of the spheres will be 12R - x.

Using the above information, we can calculate the gravitational force between the two spheres at that distance:

F = G * (m1 * m2) / (r - x)^2

Since the gravitational force is conservative, the work done by the gravitational force is given by:

W = - ∫F dx

where W is the work done and the negative sign indicates that the force is doing work against the displacement.

Integrating the above expression, we get:

W = ∫G * (m1 * m2) / (r - x)^2 dx

Simplifying the expression, we get:

W = - G * (m1 * m2) / (r - x)

To find the distance traveled by the smaller sphere, we need to equate the work done by the gravitational force to the initial kinetic energy of the smaller sphere:

W = (1/2) * m * v^2

where v is the velocity of the smaller sphere.

Equating the above expressions, we get:

(1/2) * m * v^2 = - G * (m1 * m2) / (r -