A gun is fired vertically into a 1.60 kg block of wood at rest directly above it. If the bullet has a mass of 21.0 g and a speed of 195 m/s, how high will the block rise into the air after the bullet becomes embedded in it?

initial momentum = .021 *195

final momentum = (.021+1.6)Vi
Vi = 2.53

(1/2) m Vi^2 = m g h
h = Vi^2/2g

To solve this problem, we can use the principle of conservation of momentum.

Step 1: Find the initial momentum of the bullet:
The momentum of an object is given by the product of its mass and velocity.
The mass of the bullet is 21.0 g, which is 0.021 kg. The velocity of the bullet is 195 m/s.
So, the initial momentum of the bullet is:
Momentum(bullet) = mass(bullet) * velocity(bullet)
Momentum(bullet) = 0.021 kg * 195 m/s

Step 2: Find the final velocity of the bullet and block after collision:
Since the bullet becomes embedded in the block, the final velocity of the bullet and block system will be the same.
We can use the principle of conservation of momentum to relate the initial and final momenta:
Momentum(bullet) + Momentum(block) = Total momentum after collision (which is 0 since the block comes to rest)

Step 3: Find the mass of the block:
The block has a mass of 1.60 kg according to the problem statement.

Step 4: Find the final velocity of the bullet and block:
Momentum(bullet) + Momentum(block) = 0
0.021 kg * 195 m/s + mass(block) * final velocity = 0

Solving for the final velocity of the bullet and block system, we get:
mass(block) * final velocity = -0.021 kg * 195 m/s
final velocity = -0.021 kg * 195 m/s / mass(block)

Step 5: Find the height the block rises:
To find the height the block rises, we can use the principle of conservation of mechanical energy.
The initial mechanical energy of the bullet and block system is equal to the final mechanical energy.

The initial mechanical energy is given by the sum of the potential energy and kinetic energy of the bullet:
Initial mechanical energy = potential energy(bullet) + kinetic energy(bullet)
Potential energy(bullet) = mgh
Kinetic energy(bullet) = 0.5 * mass(bullet) * velocity(bullet)^2

The final mechanical energy is given by the potential energy of the block:
Final mechanical energy = potential energy(block)
Potential energy(block) = mgh

Step 6: Equate the initial and final mechanical energy and solve for the height:
potential energy(bullet) + kinetic energy(bullet) = potential energy(block)
mgh + 0.5 * mass(bullet) * velocity(bullet)^2 = mgh

Simplifying the equation, we get:
0.5 * mass(bullet) * velocity(bullet)^2 = mgh

Solving for the height, we get:
height = (0.5 * mass(bullet) * velocity(bullet)^2) / (mass(block) * g)

Using the given values, where g is the acceleration due to gravity (9.8 m/s^2), you can substitute the values and solve for the height.

To solve this problem, we can use the principle of conservation of momentum. We can assume that there are no external forces acting on the system (bullet + block) vertically, so the total momentum of the system before and after the collision will be conserved.

Let's break down the problem into two parts:

1. Calculate the initial momentum of the bullet:
The momentum (p) of an object is given by the product of its mass (m) and velocity (v): p = m * v.
Given the mass of the bullet (m = 21.0 g = 0.021 kg) and its velocity (v = 195 m/s), calculate the initial momentum of the bullet.

Initial momentum of the bullet = mass * velocity = 0.021 kg * 195 m/s.

2. Calculate the final velocity of the bullet and the block combined:
After the bullet becomes embedded in the block, the combined mass of the bullet and the block will be (m_block + m_bullet), where the mass of the block (m_block) is 1.60 kg and the mass of the bullet is 0.021 kg.

We can calculate the final velocity (v_final) using the principle of conservation of momentum:
Initial momentum of the bullet = final momentum of the bullet and block.
(m_bullet * v_bullet)_initial = (m_block + m_bullet) * v_final

Now, let's substitute the values into the equation and solve for the final velocity.

(m_bullet * v_bullet)_initial = (m_block + m_bullet) * v_final
(0.021 kg * 195 m/s) = (1.60 kg + 0.021 kg) * v_final

Calculate the value of v_final.

Now, we have the final velocity of the bullet and the block combined. When the bullet and the block reach their highest point, their kinetic energy will be converted to potential energy (gravitational potential energy).

3. Calculate the maximum height reached by the block:
The gravitational potential energy (PE) of an object is given by the product of its mass (m) and the acceleration due to gravity (g) times the height (h): PE = m * g * h.
At the highest point, the kinetic energy of the bullet and block will be fully converted to potential energy. Therefore, the initial kinetic energy of the bullet and block is equal to the maximum potential energy they reach.

Initial kinetic energy of the bullet and block = (1/2) * (m_block + m_bullet) * v_final^2.
Maximum potential energy (gravitational potential energy) = (m_block + m_bullet) * g * h.

Now, equate the two equations and solve for h (maximum height).

(1/2) * (m_block + m_bullet) * v_final^2 = (m_block + m_bullet) * g * h

Cancel out the terms (m_block + m_bullet) from both sides, and solve for h.

Now you have the final value of h, which represents the maximum height the block rises into the air after the bullet becomes embedded in it. Substitute the given values (mass of the block, mass of the bullet, and gravitational acceleration) into the equation to obtain the numerical value of h.