To find the unknown speed of the bullet, we can use the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.
Step 1: Find the initial momentum before the collision.
The initial momentum of the bullet can be calculated using the formula:
P_initial = m_bullet * v_bullet
where:
m_bullet = mass of the bullet = 5.0 g = 0.005 kg (converted from grams to kilograms)
v_bullet = unknown speed of the bullet
P_initial = 0.005 kg * v_bullet
Step 2: Find the final momentum after the collision.
After the collision, the bullet is embedded in the block, so the final momentum is given by:
P_final = (m_bullet + m_block) * v'
where:
m_block = mass of the block = 0.195 kg
v' = final velocity of the block and bullet system
P_final = (0.005 kg + 0.195 kg) * v'
Step 3: Apply the principle of conservation of momentum.
According to the principle of conservation of momentum, the initial momentum is equal to the final momentum:
P_initial = P_final
0.005 kg * v_bullet = (0.005 kg + 0.195 kg) * v'
Simplifying the equation:
0.005 kg * v_bullet = 0.200 kg * v'
Step 4: Calculate the unknown velocity.
To find the unknown velocity v', we need to know the compression distance of the spring.
Given:
k = 180 N/m (spring constant)
x = 5.0 × 10^-2 m (compression distance)
The potential energy stored in the compressed spring can be calculated using the formula:
Potential energy = (1/2) k x^2
Step 5: Find the potential energy stored in the spring.
Potential energy = (1/2) * 180 N/m * (5.0 × 10^-2 m)^2
Step 6: Equate the potential energy to the kinetic energy of the block and bullet system.
According to the principle of conservation of mechanical energy, the potential energy stored in the spring is equal to the kinetic energy of the block and bullet system:
Potential energy = Kinetic energy
(1/2) * m_block * v'^2 = (1/2) * (m_bullet + m_block) * v'^2
Substituting the known values:
(1/2) * 0.195 kg * v'^2 = (1/2) * (0.005 kg + 0.195 kg) * v'^2
Step 7: Solve for the unknown velocity v'.
Simplifying the equation:
0.195 kg * v'^2 = 0.200 kg * v'^2
Step 8: Cancel out the squared terms and solve for v'.
Subtracting 0.200 kg * v'^2 from both sides of the equation:
0.195 kg * v'^2 - 0.200 kg * v'^2 = 0
0.005 kg * v'^2 = 0
Since the left side of the equation is equal to zero, it implies that v' should also be zero.
Therefore, the unknown velocity v' is zero, which means that the block and bullet system comes to a complete stop after the collision.
Note: This result indicates that the block and bullet system loses all its initial energy, converting it into potential energy stored in the compressed spring.