Final answer:
The initial speed of the bullet can be determined using conservation of momentum and energy, considering both the inelastic collision between the bullet and the block, and the subsequent swing to the maximum angle. After calculations, the initial speed of the bullet is found to be approximately 464.9 m/s.
Step-by-step explanation:
To calculate the initial speed of the bullet before it embeds itself into the block of wood, we can use the principles of conservation of momentum and energy. Initially, when the bullet is fired into the block, their momentum is conserved since the collision is inelastic (the bullet embeds itself in the block).
Let's denote the bullet's mass mb, the bullet's initial velocity vb, the block's mass mw, and the combined velocity after the collision v. The conservation of momentum gives us: mb * vb = (mb + mw) * v. Solving for v leads to v = mb * vb / (mb + mw).
After the collision, the block and the bullet swing together to a height h that corresponds to the angle of 12 degrees. At the highest point of the swing, all kinetic energy has been converted into potential energy (mb + mw)gh, where g is the acceleration due to gravity and h = L * (1 - cos(θ)), where L is the length of the string and θ is the angle. By conservation of energy, the kinetic energy right after the collision should equal the potential energy at the highest point of the swing. Therefore, (1/2) * (mb + mw) * v^2 = (mb + mw)gh.
After rearranging and canceling out (mb + mw), we can solve for v: v = √(2gh). Substituting this back into the momentum formula and solving for vb enables us to find the initial speed of the bullet: vb = v * (mb + mw) / mb.
With the given masses, string length, and angle, we can calculate the initial speed of the bullet to be approximately 464.9 m/s.