Final answer:
To calculate the bullet's initial speed, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy. Given the masses of the bullet and the pendulum, as well as the vertical distance the center of mass of the pendulum rises, we can calculate the initial velocity of the pendulum center of mass and then determine the bullet's initial speed.
Step-by-step explanation:
To calculate the bullet's initial speed, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy. The momentum before the collision is equal to the momentum after the collision, and the mechanical energy before the collision is equal to the mechanical energy after the collision.
Given:
Mass of the bullet (m1): 4.5 g = 0.0045 kg
Mass of the pendulum (m2): 3.0 kg
Vertical distance the center of mass of the pendulum rises (Δh): 8.1 cm = 0.081 m
Step 1: Calculate the initial velocity of the pendulum center of mass.
The mechanical energy before the collision is given by:
E1 = m2 * g * h1
Where:
E1 = mechanical energy before the collision
m2 = mass of the pendulum
g = acceleration due to gravity = 9.8 m/s2
h1 = initial height of the pendulum
Plugging in the values, we get:
E1 = 3.0 kg * 9.8 m/s2 * 0.081 m = 2.3784 J
The mechanical energy after the collision is given by:
E2 = (m1 + m2) * g * h2
Where:
E2 = mechanical energy after the collision
m1 = mass of the bullet
m2 = mass of the pendulum
g = acceleration due to gravity = 9.8 m/s2
h2 = final height of the pendulum
Since the bullet remains embedded in the pendulum, the final height (h2) is zero.
Therefore, E2 = (0.0045 kg + 3.0 kg) * 9.8 m/s2 * 0 = 0 J
Using the principle of conservation of mechanical energy, we have:
E1 = E2
2.3784 J = 0 J
Since the mechanical energy is conserved, the initial height of the pendulum (h1) is equal to the total distance the center of mass of the pendulum rises (Δh).
Therefore, h1 = 0.081 m
Step 2: Calculate the initial velocity of the pendulum center of mass.
The initial velocity (v1) can be calculated using the equation:
v1 = √(2 * g * h1)
Plugging in the values, we get:
v1 = √(2 * 9.8 m/s2 * 0.081 m) = 0.9 m/s
Step 3: Calculate the initial speed of the bullet (v0).
Since the bullet remains embedded in the pendulum, the initial speed of the bullet is equal to the initial velocity of the pendulum center of mass.
Therefore, v0 = 0.9 m/s.