A 24 rifle bullet traveling 230 buries itself in a 3.9 pendulum hanging on a 3.0 long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum's displacement.
To determine the vertical and horizontal components of the pendulum's displacement, we need to understand the initial state of the bullet-pendulum system and the principles of conservation of momentum and energy.
1. **Initial state:** Initially, the bullet is traveling horizontally with a certain velocity (let's call it v) and mass (let's call it m). The pendulum is at rest, hanging vertically with a length of 3.0 m.
2. **Conservation of momentum:** When the bullet hits the pendulum, the momentum of the system is conserved. Since the pendulum is initially at rest, the initial momentum is zero. After the collision, the combined system of the bullet and the pendulum will move together. Therefore, the momentum after the collision will also be zero.
The momentum (p) can be calculated as the product of mass (m) and velocity (v): p = m * v.
Since the momentum before and after the collision should be equal, we have:
Initial momentum (before collision) = Final momentum (after collision)
0 = (m_bullet + m_pendulum) * V_final
3. **Conservation of energy:** The potential energy (PE) of the pendulum is converted into the kinetic energy (KE) of the bullet-pendulum system. Assuming there is no energy loss due to air resistance or other factors, the energy is conserved.
The potential energy of the pendulum is given by PE = m_pendulum * g * h, where g is the acceleration due to gravity and h is the vertical displacement of the pendulum from its equilibrium position.
The kinetic energy of the bullet-pendulum system is given by KE = (m_bullet + m_pendulum) * V^2 / 2, where V is the velocity of the combined system after the collision.
Equating the potential energy before the collision with the kinetic energy after the collision, we have:
m_pendulum * g * h = (m_bullet + m_pendulum) * V_final^2 / 2
Now, let's solve these equations to find the vertical and horizontal components of the pendulum's displacement.
First, we need additional information about the bullet, such as its mass and velocity. Please provide the necessary details for further calculations.