Initial Conditions:
- Mass of the sphere: M
- Velocity of the sphere: V0
- Mass of the pendulum bob: m

Collision:
When the sphere collides with the pendulum bob, the two masses stick together and move as a single object.

Conservation of Momentum:
The total momentum before the collision is equal to the total momentum after the collision, assuming no external forces act on the system.

Calculating Initial Momentum:
The initial momentum of the system (sphere + pendulum bob) is calculated as the product of the total mass (M + m) and the initial velocity of the sphere (V0).

Calculating Final Velocity:
Since the two masses stick together after the collision, we can calculate the final velocity of the combined masses using the conservation of momentum equation. The total mass of the combined masses is M + m, and the final velocity is denoted by Vf.

Calculating Maximum Height:
The maximum height attained by the combined masses can be calculated using the principle of conservation of mechanical energy, specifically the conservation of gravitational potential energy and kinetic energy.

Conversion of Kinetic Energy to Potential Energy:
At the maximum height, all the kinetic energy of the combined masses is converted into potential energy. Therefore, we equate the initial kinetic energy (1/2)(M + m)Vf^2 to the potential energy (M + m)gh, where g is the acceleration due to gravity and h is the maximum height.

Solving for Maximum Height:
By equating the initial kinetic energy to the potential energy, we can solve for the maximum height h. Rearranging the equation gives:

h = (Vf^2) / (2g)

Substituting the expression for Vf obtained from the conservation of momentum equation yields:

h = (V0^2) / (2g)

Conclusion:
The maximum height attained by the combined masses after the collision is (V0^2) / (2g). This calculation assumes no energy losses due to friction or other dissipative forces.