A(n) 3.46 g bullet is fired into a(n) 2.4 kg ballistic pendulum and becomes embedded in it. The acceleration of gravity is 9.8 m/s2 . If the pendulum rises a vertical distance of 7.13 cm, calculate the initial speed of the

bullet.

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

1. Conservation of momentum:
The total momentum before the interaction is equal to the total momentum after the interaction. In this case, the bullet is embedded in the ballistic pendulum, so both bullet and pendulum move together after the collision.

Before the collision:
Momentum of the bullet = mass of the bullet × initial velocity of the bullet (p_bullet = m_bullet * v_bullet)

After the collision:
Total momentum of the bullet + pendulum = (mass of the bullet + mass of the pendulum) × final velocity of the bullet and pendulum (p_total = (m_bullet + m_pendulum) * v_final)

Since momentum is conserved, we can set the two expressions equal to each other:
m_bullet * v_bullet = (m_bullet + m_pendulum) * v_final

2. Conservation of energy:
The total mechanical energy before the interaction is equal to the total mechanical energy after the interaction.

Before the collision:
The bullet has kinetic energy given by the expression: KE_bullet = (1/2) * m_bullet * v_bullet^2

After the collision:
The bullet and pendulum together have potential energy given by the expression: PE_total = (m_bullet + m_pendulum) * g * h

Where g is the acceleration due to gravity and h is the vertical distance the pendulum rises.

Since energy is conserved, we can set the two expressions equal to each other:
(1/2) * m_bullet * v_bullet^2 = (m_bullet + m_pendulum) * g * h

Now, we have two equations with two unknowns (v_bullet and v_final). We can solve these equations simultaneously to find the value of v_bullet.

Substituting the value of v_final from the first equation into the second equation, we get:
(1/2) * m_bullet * v_bullet^2 = (m_bullet + m_pendulum) * g * h

Simplifying:
m_bullet * v_bullet^2 = 2 * (m_bullet + m_pendulum) * g * h

Rearranging the equation to solve for v_bullet:
v_bullet^2 = (2 * (m_bullet + m_pendulum) * g * h) / m_bullet

Taking the square root of both sides:
v_bullet = sqrt((2 * (m_bullet + m_pendulum) * g * h) / m_bullet)

Now, substituting the given values into the equation, we can calculate the initial speed of the bullet.

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

Step 1: Find the final velocity of the bullet and the pendulum after the collision.
Using the principle of conservation of momentum:
m1v1 + m2v2 = (m1 + m2)v
Here, m1 is the mass of the bullet, v1 is the initial velocity of the bullet, m2 is the mass of the pendulum, v2 is the initial velocity of the pendulum (assumed to be 0), and v is the final velocity of the bullet and pendulum after the collision.
Rearranging the equation, we get:
v = (m1v1)/(m1 + m2)

Step 2: Find the final height of the pendulum after the collision.
Using the principle of conservation of energy:
Initial kinetic energy of the system = Final potential energy of the system
(1/2)(m1 + m2)v^2 = (m1 + m2)gh
Here, g is the acceleration due to gravity, h is the final vertical distance the pendulum rises.
Rearranging the equation, we get:
v^2 = 2gh

Step 3: Substitute the values into the equations and solve for v.
Using the given values:
m1 = 3.46 g = 0.00346 kg
m2 = 2.4 kg
g = 9.8 m/s^2
h = 7.13 cm = 0.0713 m

Substituting into the equations:
v = (0.00346 kg * v1) / (0.00346 kg + 2.4 kg)
v^2 = 2 * 9.8 m/s^2 * 0.0713 m

Solving the equations simultaneously will give us the value of v.

Step 4: Calculate the initial velocity of the bullet.
Since the bullet becomes embedded in the pendulum, the final velocity of the pendulum is equal to v. Therefore, the initial velocity of the bullet (v1) can be calculated as follows:
v = m2 * v2 = m1 * v1
v1 = v * (m1 + m2) / m1

Substituting the calculated value of v and the given values of m1 and m2 will give us the initial velocity of the bullet.