To solve this problem, we can make use of the principle of conservation of linear momentum and the principle of conservation of mechanical energy.
(a) To find the initial speed of the bullet, we can use the principle of conservation of linear momentum. This principle states that the total momentum before the collision is equal to the total momentum after the collision.
Let the initial speed of the bullet be v and the final velocity of the bullet-bob combination immediately after the collision be V.
The momentum of the bullet before the collision is given by the formula:
Momentum of bullet = mass of bullet * initial velocity of bullet
= m * v
The momentum of the bob immediately after the collision is given by the formula:
Momentum of bob = mass of bob * final velocity of the bullet-bob combination
= M * V
According to the principle of conservation of linear momentum:
Momentum of bullet = Momentum of bob
m * v = M * V
Now, let's find the final velocity of the bullet-bob combination, V.
We can use the principle of conservation of mechanical energy to relate the initial and final velocities of the bullet-bob combination.
The initial kinetic energy of the bullet-bob combination is given by the formula:
Initial kinetic energy = (1/2) * (mass of bullet + mass of bob) * (initial velocity)^2
The final potential energy of the bullet-bob combination is given by the formula:
Final potential energy = (mass of bob) * (acceleration due to gravity) * (height bob rises)
According to the principle of conservation of mechanical energy:
Initial kinetic energy = Final potential energy
(1/2) * (m + M) * (v^2) = M * g * h
Where g is the acceleration due to gravity and h is the height the bob rises.
Now, substitute v with V in the equation obtained from the principle of conservation of linear momentum:
m * v = M * V
Solve these two equations simultaneously to find the values of V and v.
(b) Once we have found the value of V, we can calculate the speed of the bullet-bob combination immediately after the collision by simply taking the absolute value of V, as speed is the magnitude of velocity.