The initial speed of the bullet was approximately 28.03 m/s.
1. Conservation of momentum:
Since the pendulum bob is initially at rest, the total momentum before the collision is just the momentum of the bullet:
P_initial = P_bullet = mv_0
After the collision, the bullet and bob become a single combined object, and its momentum is conserved:
P_final = (m_bullet + m_bob) * v_f
where:
m_bullet = 8.00 g = 0.00800 kg
m_bob = 1.20 kg
v_f is the velocity of the combined object after the collision
Since the bullet embeds itself in the bob, their velocities become equal after the collision:
v_f = v_final (combined object)
Therefore, the conservation of momentum equation becomes:
mv_0 = (m_bullet + m_bob) * v_final
2. Conservation of mechanical energy:
At the highest point of the swing, the combined object's kinetic energy is converted entirely to potential energy.
E_initial = E_final
(1/2) * (m_bullet + m_bob) * v_final^2 = (m_bullet + m_bob) * g * h
where:
g = 9.81 m/s^2 (acceleration due to gravity)
h = 15.0 cm = 0.150 m (maximum height reached)
3. Solve the equations:
From the conservation of momentum equation:
v_final = mv_0 / (m_bullet + m_bob)
Substitute this expression into the conservation of energy equation:
(1/2) * (m_bullet + m_bob) * (mv_0 / (m_bullet + m_bob))^2 = (m_bullet + m_bob) * g * h
Simplify and solve for v_0:
(1/2) * m * v_0^2 = (m_bullet + m_bob) * g * h
v_0 = sqrt(2 * (m_bullet + m_bob) * g * h / m)
Plug in the values:
v_0 = sqrt(2 * (0.00800 kg + 1.20 kg) * 9.81 m/s^2 * 0.150 m / 0.00800 kg)
v_0 ≈ 28.03 m/s
Therefore, the initial speed of the bullet was approximately 28.03 m/s.