To find the speed of the bullet at the moment it impacts the block, we can use the principles of conservation of momentum and energy. When the bullet embeds itself in the block, the system conserves momentum, and the kinetic energy of the bullet-block system is converted into potential energy stored in the spring as it compresses. Let's break this down step by step.
Step 1: Understanding the System
We have a bullet with a mass of 12.0 g (which we convert to kilograms for our calculations, so 0.012 kg) and a wooden block with a mass of 100 g (or 0.1 kg). The block is initially at rest, and the bullet is fired horizontally. When the bullet embeds itself in the block, they move together as one mass.
Step 2: Conservation of Momentum
The principle of conservation of momentum states that the total momentum before the collision must equal the total momentum after the collision. Before the collision, only the bullet has momentum, while the block is stationary. After the collision, the bullet and block move together.
The equation for momentum is:
- Initial momentum = Final momentum
- m_bullet * v_bullet = (m_bullet + m_block) * v_final
Here, m_bullet is the mass of the bullet, v_bullet is the speed of the bullet before impact, and v_final is the speed of the bullet-block system immediately after the collision.
Step 3: Energy Considerations
After the bullet embeds in the block, the kinetic energy of the bullet-block system is converted into potential energy stored in the spring when it is compressed. The potential energy (PE) in the spring can be expressed as:
Where k is the spring constant (150 N/m) and x is the maximum compression of the spring (0.8 m). The kinetic energy (KE) of the bullet-block system just after the collision is:
- KE = (1/2) * (m_bullet + m_block) * v_final²
Step 4: Setting Up the Equations
From the conservation of momentum, we can express v_final as:
- v_final = (m_bullet * v_bullet) / (m_bullet + m_block)
Now, we can equate the kinetic energy just after the collision to the potential energy at maximum compression:
- (1/2) * (m_bullet + m_block) * v_final² = (1/2) * k * x²
Step 5: Solving for the Bullet's Speed
Substituting the expression for v_final into the energy equation gives us:
- (1/2) * (m_bullet + m_block) * [(m_bullet * v_bullet) / (m_bullet + m_block)]² = (1/2) * k * x²
We can simplify this equation and solve for v_bullet. First, cancel the (1/2) on both sides:
- (m_bullet + m_block) * [(m_bullet * v_bullet)² / (m_bullet + m_block)²] = k * x²
Now, rearranging gives:
- v_bullet² = (k * x² * (m_bullet + m_block)) / m_bullet²
Substituting the known values:
- k = 150 N/m
- x = 0.8 m
- m_bullet = 0.012 kg
- m_block = 0.1 kg
Calculating:
- v_bullet² = (150 * (0.8)² * (0.012 + 0.1)) / (0.012)²
- v_bullet² = (150 * 0.64 * 0.112) / 0.000144
- v_bullet² = (10.08) / 0.000144
- v_bullet² = 70000
- v_bullet = √70000 ≈ 264.57 m/s
Final Result
The speed of the bullet at impact with the block is approximately 264.57 m/s. This calculation illustrates how momentum and energy conservation principles work together in a collision scenario.
