By conservation of momentum principle the final velocity right after collision is v=100/2 = 50cm/sec
Hence by conservation of energy theorem ; 1/2mv²=mgh
So , h=10×21​0.52
=1.25m

Analysis:
In order to determine the maximum height reached by the system after the collision, we need to consider the principles of conservation of momentum and conservation of energy.

Conservation of Momentum:
The principle of conservation of momentum states that the total momentum of a system before a collision is equal to the total momentum after the collision, provided no external forces are acting on the system. Mathematically, this can be expressed as:

Initial momentum = Final momentum

Since the mass of the pendulum bob is the same as the mass of the moving mass, the initial momentum can be calculated as:

Initial momentum = mass of the moving mass × velocity of the moving mass

Given that the mass of the moving mass is 10 gm (0.01 kg) and the velocity is 100 cm/sec (1 m/sec), the initial momentum can be calculated as:

Initial momentum = 0.01 kg × 1 m/sec = 0.01 kg·m/sec

After the collision, both masses stick together and move as a single system. Let's denote the final velocity of the system as V.

Final momentum = (mass of the moving mass + mass of the pendulum bob) × final velocity

Since both masses are the same, the final momentum can be calculated as:

Final momentum = 2 × mass of the moving mass × final velocity

According to the principle of conservation of momentum, the initial momentum is equal to the final momentum:

0.01 kg·m/sec = 2 × 0.01 kg × V

Simplifying the equation, we find that the final velocity of the system is:

V = 0.01 m/sec

Conservation of Energy:
The principle of conservation of energy states that the total energy of a system remains constant if no external work is done on the system. In this case, the initial energy of the system is the kinetic energy of the moving mass, and the final energy of the system is the potential energy at the maximum height reached.

The initial kinetic energy can be calculated using the formula:

Initial kinetic energy = 0.5 × mass of the moving mass × (velocity of the moving mass)^2

Substituting the given values, we find:

Initial kinetic energy = 0.5 × 0.01 kg × (1 m/sec)^2 = 0.005 J

At the maximum height, all the initial kinetic energy is converted into potential energy. Therefore, the potential energy at the maximum height can be calculated as:

Potential energy = Initial kinetic energy = 0.005 J

The potential energy can also be expressed as:

Potential energy = mass of the system × acceleration due to gravity × height

Since the mass of the system is 0.02 kg (sum of the masses of the moving mass and the pendulum bob) and the acceleration due to gravity is 9.8 m/s^2, we can calculate the height as:

Height = Potential energy / (mass of the system × acceleration due to gravity)

Height = 0.005 J / (0.02 kg × 9.8 m/s^2) = 0.025 m = 2.5 cm

Therefore, the maximum height reached by the system after the collision is 2.5 cm.