An 8.82g bullet is fired into a 235g block that is initially at rest at the edge of a table of h = 1.05 m height (see the figure below).


The bullet remains in the block, and after the impact the block lands d = 2.16 m from the bottom of the table. Determine the initial speed of the bullet.

To determine the initial speed of the bullet, we can use the principle of conservation of momentum. The principle states that the total momentum of an isolated system remains constant before and after a collision.

In this case, the system consists of the bullet and the block. Before the collision, both the bullet and the block are at rest, so the total initial momentum is zero. After the collision, the bullet and the block move together and land 2.16 m from the bottom of the table.

The momentum of an object is calculated by multiplying its mass by its velocity. Let's assume the initial velocity of the bullet is denoted as v and the mass of the block is m. The mass of the bullet is given as 8.82 g, which is equal to 0.00882 kg.

Initially, the momentum of the system is zero:
0 = m * 0 + 0.00882 kg * v

After the collision, the bullet and the block move together. Let's denote their combined mass as M, which is equal to the sum of the bullet's mass and the block's mass (m + 0.235 kg). We need to determine the velocity at which the bullet and the block move together.

Using the principle of conservation of momentum, the total momentum after the impact can be determined as:
(M) * (velocity) = M * (final velocity)

When the block and bullet land on the ground, their combined mass (M) is given as 0.235 kg + 0.00882 kg. The final velocity can be calculated using the equation:
final velocity = (2.16 m) / (time taken to fall)

To calculate the time taken to fall, we can use the equation for the free-fall time from height h:
time taken to fall = sqrt((2 * h) / g)

where g is the acceleration due to gravity (approximately 9.8 m/s^2). Plugging in the values, we have:
time taken to fall = sqrt((2 * 1.05 m) / 9.8 m/s^2)

Now, we can substitute the values back into the equation for momentum conservation:
(M) * (velocity) = (M) * (final velocity)

Simplifying the equation, we find:
0.24382 kg * v = (0.24382 kg + 0.00882 kg) * (2.16 m) / sqrt((2 * 1.05 m) / 9.8 m/s^2)

Solving this equation will give us the initial velocity (v) of the bullet.