A 1.75-kg wooden block rests on a table over a large hole as in the figure below. A 4.60-g bullet with an initial velocity vi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of 24.0 cm.

Calculate the initial velocity of the bullet from the information provided. (Let up be the positive direction.)

To calculate the initial velocity of the bullet, we can use the principle of conservation of energy. In this problem, the initial kinetic energy of the bullet is transformed into potential energy when the block and bullet reach their maximum height.

We first need to calculate the potential energy of the block and bullet at their maximum height using the formula:

PE = m * g * h

Where:
PE = potential energy
m = total mass of the block and the bullet
g = acceleration due to gravity (approximately 9.81 m/s²)
h = maximum height reached by the block and the bullet

Given that the mass of the block is 1.75 kg and the height reached is 24.0 cm (0.24 m), we can calculate the potential energy:

PE = (1.75 kg + 0.00460 kg) * 9.81 m/s² * 0.24 m

Next, we need to calculate the initial kinetic energy of the bullet. This can be done using the formula:

KE = (1/2) * m * v²

Where:
KE = initial kinetic energy of the bullet
m = mass of the bullet (0.00460 kg)
v = initial velocity of the bullet (to be calculated)

Since the bullet is fired upward, its change in height is the same as the height reached by the block and the bullet:
Δh = h = 0.24 m

The potential energy at the maximum height can be assumed to be equal to the initial kinetic energy of the bullet, as energy is conserved:

PE = KE

Setting the two equations equal to each other:

(1.75 kg + 0.00460 kg) * 9.81 m/s² * 0.24 m = (1/2) * 0.00460 kg * v²

Now, we can solve for v:

Simplify the equation:

(1.75460 kg) * 9.81 m/s² * 0.24 m = (1/2) * 0.00460 kg * v²

Multiply both sides by 2 to get rid of the fraction:

(1.75460 kg) * 9.81 m/s² * 0.24 m * 2 = 0.00460 kg * v²

Rearranging the equation:

v² = [(1.75460 kg) * 9.81 m/s² * 0.24 m * 2] / 0.00460 kg

Taking the square root of both sides to solve for v:

v = √{[(1.75460 kg) * 9.81 m/s² * 0.24 m * 2] / 0.00460 kg}