Problem:
A block of mass 10g travelling at a speed of 500 m/s strikes a block of mass 2 kg, which is suspended by a string of length 5m. The center of gravity of the block is found to rise a vertical distance of 0.2m. What is the speed of the bullet after it emerges from the block?

Solution:
To find the speed of the bullet after it emerges from the block, we need to consider the principle of conservation of momentum and energy.

1. Conservation of Momentum:
When the bullet strikes the block, momentum is conserved. The initial momentum of the bullet is given by:
P_initial = mass_bullet * velocity_bullet

The momentum of the bullet after exiting the block is given by:
P_final = mass_bullet * velocity_bullet_final

Since momentum is conserved, we have:
P_initial = P_final

Substituting the values, we have:
mass_bullet * velocity_bullet = mass_bullet * velocity_bullet_final

2. Conservation of Energy:
When the bullet strikes the block, some of its kinetic energy is used to raise the center of gravity of the block. We can calculate the work done by the bullet on the block using the work-energy principle:

Work_done = change_in_potential_energy

The work done by the bullet is given by:
Work_done = force * distance

The force can be calculated using Newton's second law:
Force = mass_block * acceleration

The acceleration can be calculated using the equation of motion:
distance = (1/2) * acceleration * time^2

Simplifying these equations, we have:
Work_done = (mass_block * acceleration) * distance
Work_done = (mass_block * acceleration) * (1/2) * (time^2)

The change in potential energy is given by:
change_in_potential_energy = mass_block * gravity * change_in_height

Since the center of gravity rises a vertical distance of 0.2m, we have:
change_in_potential_energy = mass_block * gravity * 0.2

Equating the work done and change in potential energy, we have:
(mass_block * acceleration) * (1/2) * (time^2) = mass_block * gravity * 0.2

3. Solving for Time:
From the equation of motion, we know that:
distance = velocity_bullet * time

Substituting the given values, we have:
5 = 500 * time

Solving for time, we find:
time = 5/500 = 0.01s

4. Solving for Velocity of Bullet After Emerging:
From the equation of conservation of momentum, we have:
mass_bullet * velocity_bullet = mass_bullet * velocity_bullet_final

Simplifying, we find:
velocity_bullet_final = velocity_bullet

Therefore, the speed of the bullet after it emerges from the block is 500 m/s.