This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Identify the given values for the masses and initial velocities. Mass of the lead bullet (m_1) = 0.05 \, kg Initial velocity of the lead bullet (v_1) = 200 \, m/s Mass of the lead block (m_2) = 0.95 \, kg Initial velocity of the lead block (v_2) = 0 \, m/s (since it is at rest) Step 2: Recognize that this is a perfectly inelastic collision. When the bullet is fired into the block and the block can move freely, it implies that the bullet embeds itself in the block, and they move together as a single combined mass after the impact. In such a collision, momentum is conserved, but kinetic energy is not. Step 3: Apply the principle of conservation of momentum to find the final velocity of the combined mass. Let V_f be the final velocity of the combined bullet-block system. The total momentum before the collision equals the total momentum after the collision: m_1 v_1 + m_2 v_2 = (m_1 + m_2) V_f Substitute the given values: (0.05 \, kg)(200 \, m/s) + (0.95 \, kg)(0 \, m/s) = (0.05 \, kg + 0.95 \, kg) V_f 10 \, kg · m/s + 0 = (1.00 \, kg) V_f 10 = 1.00 V_f V_f = (10)/(1.00) \, m/s V_f = 10 \, m/s Step 4: Calculate the final kinetic energy of the combined system. The final kinetic energy (KE_f) is given by the formula KE = (1)/(2) M V^2, where M is the total mass and V is the final velocity. KE_f = (1)/(2) (m_1 + m_2) V_f^2 Substitute the total mass and the final velocity: KE_f = (1)/(2) (1.00 \, kg) (10 \, m/s)^2 KE_f = (1)/(2) (1.00 \, kg) (100 \, m^2/s^2) KE_f = 0.5 × 100 \, J KE_f = 50 \, J The final kinetic energy after impact is 50 J.