To solve the problem, we need to determine the final kinetic energy of a lead bullet after it collides with a lead block. We will use the principles of conservation of momentum and kinetic energy. ### Step 1: Understand the Problem We have: - A lead bullet with mass \( m_1 = 0.05 \, \text{kg} \) and initial velocity \( v_1 = 200 \, \text{m/s} \). - A lead block with mass \( m_2 = 0.95 \, \text{kg} \) that is initially at rest, so its initial velocity \( v_2 = 0 \, \text{m/s} \). ### Step 2: Calculate Initial Momentum The total initial momentum \( p_{\text{initial}} \) of the system (bullet + block) can be calculated using the formula: \[ p_{\text{initial}} = m_1 v_1 + m_2 v_2 \] Substituting the values: \[ p_{\text{initial}} = (0.05 \, \text{kg} \times 200 \, \text{m/s}) + (0.95 \, \text{kg} \times 0 \, \text{m/s}) = 10 \, \text{kg m/s} \] ### Step 3: Apply Conservation of Momentum After the collision, let \( v_f \) be the final velocity of both the bullet and the block (since they move together after the collision). According to the conservation of momentum: \[ p_{\text{initial}} = p_{\text{final}} \] Thus, \[ 10 \, \text{kg m/s} = (m_1 + m_2) v_f \] Substituting the masses: \[ 10 \, \text{kg m/s} = (0.05 \, \text{kg} + 0.95 \, \text{kg}) v_f \] \[ 10 \, \text{kg m/s} = 1.0 \, \text{kg} \cdot v_f \] Solving for \( v_f \): \[ v_f = \frac{10 \, \text{kg m/s}}{1.0 \, \text{kg}} = 10 \, \text{m/s} \] ### Step 4: Calculate Final Kinetic Energy The final kinetic energy \( KE_f \) of the combined system (bullet + block) can be calculated using the formula: \[ KE_f = \frac{1}{2} (m_1 + m_2) v_f^2 \] Substituting the values: \[ KE_f = \frac{1}{2} (1.0 \, \text{kg}) (10 \, \text{m/s})^2 \] \[ KE_f = \frac{1}{2} (1.0) (100) = 50 \, \text{J} \] ### Conclusion The final kinetic energy after the impact is **50 J**. Therefore, the correct option is **A**. ### Explanation of Other Options - **B. 100 J**: This would imply that the final kinetic energy is double what we calculated. This could happen if the system had more energy input or if the masses were different, but based on the conservation of momentum and the given masses, this is incorrect. - **C. 150 J**: Similar reasoning applies here; this value does not correspond to the calculated kinetic energy based on the conservation laws. - **D. 200 J**: This option suggests that the kinetic energy is equal to the initial kinetic energy of the bullet alone, which is not possible since some energy is lost in the collision (assuming it is inelastic). ### Revision Summary - Use conservation of momentum to find the final velocity after a collision. - Calculate the final kinetic energy using the combined mass and final velocity. - Ensure to account for the initial conditions of both objects involved in the collision. - Remember that in inelastic collisions, kinetic energy is not conserved, but momentum is.