A(n) 5.52 g bullet is fired into a(n) 3.07 kg ballistic pendulum and becomes embedded in it.The acceleration of gravity is 9.8 m/s^2. If the pendulum rises a vertical distance of 4.48 cm, calculate the initial speed of the bullet. Answer in units of m/s

Oh, I'm always a "ballistic" when it comes to physics questions! Let's solve this puzzle, shall we?

First, let's find the momentum of the bullet before it collides with the pendulum. The momentum (p) is given by the equation p = m × v, where m is the mass and v is the velocity.

Since the bullet becomes embedded in the pendulum, their masses combine. So we have:

(5.52 g + 3.07 kg) × v = total momentum

Converting to SI units, we get:

(0.00552 kg + 3.07 kg) × v = total momentum

Now, let's consider the conservation of momentum. Since there are no external forces acting on the system, the total momentum before the collision should be equal to the total momentum after the collision.

The total momentum after the collision is given by:

(total mass of pendulum + bullet) × final velocity

Assuming the pendulum swings up perfectly vertically, we can use the conservation of energy to relate the final velocity (vf) to the vertical distance (d) it rises:

m × g × d = (1/2) × (total mass of pendulum + bullet) × vf^2

Now it's time to put on our thinking caps and solve this puzzle! Rearranging the equation, we find:

vf = sqrt((2 × m × g × d) / (total mass of pendulum + bullet))

Plugging in the given numbers:

vf = sqrt((2 × 0.00552 kg × 9.8 m/s^2 × 0.0448 m) / (3.07 kg + 0.00552 kg))

vf = sqrt(0.0970512 / 3.07552)

vf ≈ sqrt(0.03161)

vf ≈ 0.1779 m/s

So, the final velocity after the collision is approximately 0.1779 m/s.

Since momentum is conserved, the initial momentum of the bullet is equal to the total momentum:

(5.52 g) × v = (0.00552 kg + 3.07 kg) × 0.1779 m/s

v = (3.07552 kg × 0.1779 m/s) / 0.00552 kg

v ≈ 98.87 m/s

Voilà! The initial speed of the bullet is approximately 98.87 m/s.

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

1. Conservation of momentum:
Before the bullet is fired into the pendulum, there is no initial horizontal velocity, so the momentum of the system is zero. After the bullet is embedded in the pendulum, both the pendulum and the bullet move together. Therefore, the momentum after the collision is also zero.

The momentum of the bullet can be calculated using the equation:
momentum = mass × velocity

Given:
mass of the bullet (m₁) = 5.52 g = 0.00552 kg
mass of the pendulum (m₂) = 3.07 kg

Let the velocity of the bullet be v (initial speed).

The momentum of the bullet before the collision is:
momentum_before = m₁ × 0 = 0

The momentum of the bullet after the collision is:
momentum_after = (m₁ + m₂) × v

Since momentum is conserved, we can equate the before and after momenta:
0 = (m₁ + m₂) × v

Therefore, we can solve for v:
v = 0 / (m₁ + m₂)
v = 0 m/s

2. Conservation of energy:
When the bullet is embedded in the pendulum, the system's initial mechanical energy is transformed into potential energy.

The potential energy gained by the pendulum can be calculated using the equation:
potential energy = mass × gravitational acceleration × height

Given:
mass of the pendulum (m₂) = 3.07 kg
gravitational acceleration (g) = 9.8 m/s²
height (h) = 4.48 cm = 0.0448 m

Substituting these values, we get:
potential energy = m₂ × g × h
potential energy = 3.07 kg × 9.8 m/s² × 0.0448 m

Now, since the mechanical energy is conserved, we can equate the initial kinetic energy (0.5 × m₁ × v²) to the potential energy gained.

0.5 × m₁ × v² = m₂ × g × h

Substituting the values of m₁, v, m₂, g, and h, we can solve for v:

0.5 × 0.00552 kg × v² = 3.07 kg × 9.8 m/s² × 0.0448 m

Simplifying the equation:

0.00276 kg × v² = 0.03352384 kg⋅m²/s²

Now, solving for v:

v² = 0.03352384 kg⋅m²/s² / 0.00276 kg
v² = 12.1427536 m²/s²

Finally, taking the square root of both sides to find v:

v = √(12.1427536 m²/s²)
v ≈ 3.48 m/s

Therefore, the initial speed of the bullet is approximately 3.48 m/s.

To calculate the initial speed of the bullet, we can use the principle of conservation of momentum. In this case, the momentum before the collision is equal to the momentum after the collision.

The momentum before the collision is given by the formula:
momentum = mass x velocity

The mass of the bullet is 5.52 g, which is equal to 0.00552 kg. Let's call the initial velocity of the bullet 'v'.

So, the momentum before the collision is:
momentum_before = (0.00552 kg) x (v)

After the bullet is embedded in the pendulum, the pendulum swings up to a vertical height of 4.48 cm, which is equal to 0.0448 m. This vertical distance is the potential energy gained by the pendulum.

The potential energy gained by the pendulum is given by the formula:
potential energy = mass x acceleration due to gravity x height

The mass of the pendulum is 3.07 kg, and the acceleration due to gravity is 9.8 m/s².

So, the potential energy gained by the pendulum is:
potential energy = (3.07 kg) x (9.8 m/s²) x (0.0448 m)

According to the conservation of momentum principle, the momentum before the collision is equal to the momentum after the collision. This can be expressed as:
momentum_before = momentum_after

The momentum after the collision is the momentum of the bullet and the pendulum combined. The combined mass is the mass of the bullet plus the mass of the pendulum.

So, the momentum after the collision is:
momentum_after = (5.52 g + 3.07 kg) x final_velocity

The final velocity is the velocity of the bullet and the pendulum together right after the collision.

Now we can set up the equation:
momentum_before = momentum_after

Therefore:
(0.00552 kg) x (v) = (5.52 g + 3.07 kg) x final_velocity

Now we need to solve for 'v', the initial velocity of the bullet.

Step 1: Convert all units to SI units
- Convert 5.52 g to kg ⇒ 0.00552 kg
- Convert 5.52 g to kg ⇒ 0.00552 kg

Step 2: Rearrange the equation and solve for 'v'
v = ((5.52 g + 3.07 kg) x final_velocity) / (0.00552 kg)

Step 3: Calculate the final velocity of the pendulum
Since the bullet and the pendulum are embedded together, their final velocity after the collision is the same.
The final velocity of the pendulum can be calculated using the formula for potential energy and converting it to kinetic energy.

potential energy = kinetic energy
(3.07 kg) x (9.8 m/s²) x (0.0448 m) = (0.5) x (3.07 kg + 0.00552 kg) x (final_velocity)²

Solve for final_velocity

Step 1: Simplify the equation
(3.07 kg) x (9.8 m/s²) x (0.0448 m) = (0.5) x (3.07552 kg) x (final_velocity)²

Step 2: Rearrange the equation and solve for final_velocity
final_velocity = √((2 x (3.07 kg) x (9.8 m/s²) x (0.0448 m)) / (3.07552 kg))

Step 3: Calculate final_velocity
final_velocity = √(0.8520296) = 0.9228859 m/s

Now that we have the final velocity, let's substitute it back into the equation to find the initial velocity, 'v'.

v = ((5.52 g + 3.07 kg) x final_velocity) / (0.00552 kg)
v = ((0.00552 kg + 3.07 kg) x 0.9228859 m/s) / (0.00552 kg)

v ≈ 91.9409 m/s

Therefore, the initial speed of the bullet is approximately 91.9409 m/s.

Apply conservation of energy during the upward swing to compute the velocity of the pendulm with imbedded bullet.

V= sqrt(2gH)= 0.937 m/s

Then apply conservatuion of momentum to the bullet-imbedding process to compute the initial velocity of the bullet, v.

v*m = (M+m)V

m = bullet mass.
M = pendulum mass

Solve for v.