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. According to this principle, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces are involved.

The momentum of an object is given by the product of its mass and velocity. Therefore, we can calculate the initial momentum of the bullet-block system before the collision by finding the momentum of the bullet and the momentum of the block separately.

The momentum of the bullet is given by the product of its mass and velocity, which can be represented as:

Momentum of the bullet = mass of the bullet × velocity of the bullet (equation 1)

Similarly, the momentum of the block is given by the product of its mass and velocity, which can be represented as:

Momentum of the block = mass of the block × velocity of the block (equation 2)

Since the block is initially at rest, its initial velocity is zero. Therefore, the momentum of the block before the collision is zero.

Now, after the impact, the bullet remains in the block, and the combined mass of the bullet and the block becomes the mass of the block in equation 2.

Momentum of the block after the collision = (mass of the bullet + mass of the block) × velocity of the block (equation 3)

According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:

Momentum before = Momentum after

Using equations 1, 2, and 3, we can write:

mass of the bullet × velocity of the bullet = (mass of the bullet + mass of the block) × velocity of the block

Now let's substitute the given values into the equation:

mass of the bullet = 8.82 g = 0.00882 kg
mass of the block = 235 g = 0.235 kg
height of the table, h = 1.05 m
distance from the bottom of the table, d = 2.16 m

Substituting the values into the equation, we get:

0.00882 kg × velocity of the bullet = (0.00882 kg + 0.235 kg) × velocity of the block

Simplifying the equation, we get:

velocity of the bullet = (0.24482 kg × velocity of the block) / 0.00882 kg

Now, we need to find the velocity of the block.

To find the velocity of the block, we can use the equation of motion for free fall:

d = (1/2) × g × t^2

where d is the distance fallen, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time of fall.

Rearranging the equation, we get:

t = sqrt((2 × d) / g)

Substituting the given values, we have:

t = sqrt((2 × 2.16 m) / 9.8 m/s^2)

Evaluating the expression, t ≈ 0.47 s

Since the block falls a vertical distance of h = 1.05 m, we can find the velocity of the block using the equation:

velocity of the block = g × t

Substituting the values, we get:

velocity of the block = 9.8 m/s^2 × 0.47 s

Evaluating the expression, velocity of the block ≈ 4.61 m/s

Now, we can substitute the velocity of the block into the equation for the velocity of the bullet:

velocity of the bullet = (0.24482 kg × 4.61 m/s) / 0.00882 kg

Evaluating the expression, we get:

velocity of the bullet ≈ 128 m/s

Therefore, the initial speed of the bullet is approximately 128 m/s.