A 18 g bullet is fired into the bob of a ballistic pendulum of mass 1.4 kg. When the bob is at its maximum height, the strings make an angle of 60° with the vertical. The length of the pendulum is 2.3 m. Find the speed of the bullet.

To find the speed of the bullet, we can use the principle of conservation of momentum and conservation of energy.

First, let's analyze the system at the initial and final states.

Initial state:
- The bullet is moving with some initial speed.
- The bob of the ballistic pendulum is at rest.

Final state:
- The bullet is embedded in the bob of the ballistic pendulum.
- The bob has reached its maximum height after swinging upward.

Now, let's calculate the speed of the bullet using the principle of conservation of energy.

The initial kinetic energy of the bullet is given by:
KE_initial = (1/2) * mass_bullet * velocity_bullet^2

The final kinetic energy of the bullet and the bob at the maximum height is zero since they come to rest.

The initial gravitational potential energy of the bob is converted into the kinetic energy of the bullet and the bob at their maximum height.

At the maximum height, the gravitational potential energy of the bob is given by:
PE_gravitational = mass_bob * gravitational_acceleration * height_max

Using the conservation of energy, we can equate the initial kinetic energy of the bullet to the final gravitational potential energy of the bob.

(1/2) * mass_bullet * velocity_bullet^2 = mass_bob * gravitational_acceleration * height_max

You have provided the mass of the bullet (18 g), the mass of the bob (1.4 kg), the angle formed by the strings with the vertical (60°), and the length of the pendulum (2.3 m).

To find the gravitational acceleration, we can use the formula:
gravitational_acceleration = 9.8 m/s^2

To find the maximum height (height_max) reached by the bob, we can use trigonometry.
The vertical component of the displacement of the bob is:
vertical_displacement = length_pendulum * sin(angle)

Substituting the given values into the equations, you can calculate the speed of the bullet.

To find the speed of the bullet, we can start by using the concept of conservation of momentum.

Step 1: Calculate the initial momentum of the bullet.
The initial momentum of the bullet can be calculated using the formula:
Initial momentum = mass of the bullet × initial velocity of the bullet

Given:
Mass of the bullet = 18 g = 0.018 kg
Let's assume the initial velocity of the bullet is v.

Initial momentum = 0.018 kg × v = 0.018v

Step 2: Calculate the momentum of the pendulum-bullet system after the collision.
The final momentum of the pendulum-bullet system can be calculated using the formula:
Final momentum = (mass of the bullet + mass of the pendulum) × final velocity

Given:
Mass of the pendulum = 1.4 kg
Final velocity of the pendulum-bullet system can be calculated using the concept of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

Before the collision, the bullet has momentum in the horizontal direction, while the pendulum has momentum in the vertical direction.

Since the bullet is fired horizontally, its momentum in the horizontal direction is zero. Therefore, the total momentum before the collision is equal to the momentum of the pendulum, which is in the vertical direction.

Total momentum before collision = mass of pendulum × velocity of pendulum

Let's assume the velocity of the pendulum is v'.

Total momentum before collision = 1.4 kg × v'

After the collision, the bullet sticks to the pendulum, and hence they move together.

Total momentum after collision = (mass of bullet + mass of pendulum) × final velocity

Total momentum after collision = (0.018 kg + 1.4 kg) × final velocity
Total momentum after collision = 1.418 kg × final velocity

Since total momentum before collision = total momentum after collision, we can write:
1.4 kg × v' = 1.418 kg × final velocity
v' = 1.418/1.4 × final velocity
v' = 1.012 × final velocity

Step 3: Calculate the final velocity of the pendulum and bullet system.
The final velocity of the pendulum and bullet system can be calculated using the concept of conservation of mechanical energy, which states that the initial mechanical energy is equal to the final mechanical energy.

The initial mechanical energy consists of the kinetic energy of the bullet and the potential energy of the pendulum due to its vertical height.

The final mechanical energy consists of the kinetic energy of the pendulum-bullet system at the maximum height, which is all in the form of potential energy.

Since the bullet is stuck in the pendulum, the entire kinetic energy of the bullet is converted into potential energy of the pendulum at its maximum height. Thus, we can equate the bullet's kinetic energy to the potential energy of the pendulum.

Kinetic energy of bullet = Potential energy of pendulum at maximum height

The kinetic energy formula for the bullet can be written as:
Kinetic energy = 0.5 × mass of bullet × (velocity)^2

The potential energy of the pendulum at maximum height can be expressed as:
Potential energy = mass of pendulum × g × height

Given:
Height = length of the pendulum × (1 - cos(angle))
Length of the pendulum = 2.3 m
Angle = 60°
Mass of pendulum = 1.4 kg
Acceleration due to gravity, g = 9.8 m/s^2

Therefore:
Height = 2.3 × (1 - cos(60°))
Height = 2.3 × (1 - 0.5)
Height = 2.3 × 0.5
Height = 1.15 m

The kinetic energy of the bullet is equal to the potential energy of the pendulum, so:
0.5 × 0.018 kg × v^2 = 1.4 kg × 9.8 m/s^2 × 1.15 m

Simplifying:
0.009 v^2 = 1.4 × 9.8 × 1.15

Solving for v^2:
v^2 = (1.4 × 9.8 × 1.15)/0.009

Taking the square root of both sides:
v = sqrt((1.4 × 9.8 × 1.15)/0.009)

Calculating the value of v will give us the speed of the bullet.