Final answer:
The initial speed of the bullet is found using the conservation of energy to calculate the kinetic energy just after the bullet embeds itself in the block and then equating it to the potential energy of the spring at maximum compression. Momentum conservation is also used since the block is initially at rest to relate the bullet's initial momentum to the combined system's momentum after the collision.
Step-by-step explanation:
The scenario described in the question involves a bullet embedding itself in a block that is attached to a spring. To find the initial speed of the bullet, we can use the principle of conservation of energy. The maximum compression of the spring indicates the maximum potential energy stored in the spring, which is equal to the kinetic energy of the block-bullet system just after the collision, assuming no energy is lost.
First, we calculate the spring potential energy at maximum compression using the formula:
U = (1/2) k x^2
Where U is the potential energy, k is the spring constant (805 N/m), and x is the compression distance (5.58 cm or 0.0558 m).
Then, with the potential energy calculated, we can equate it to the kinetic energy just after the collision:
(1/2) k x^2 = (1/2) m v^2
Where m is the combined mass of the bullet and block, and v is their velocity just after the collision.
To find the initial speed of the bullet, use the conservation of momentum since the block is initially at rest. Right before the collision, the system's total momentum must equal the system's total momentum just after the collision:
m_bullet * v_bullet_initial = (m_block + m_bullet) * v
We can solve this equation for v_bullet_initial, which gives us the bullet's initial speed, using the combined mass and the calculated velocity from the energy equations.
The step-by-step explanation demonstrates the application of energy conservation and momentum conservation principles to deduce the initial velocity of the bullet.