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: Define the initial and final states of the system. Let the initial mass of the object be M. The initial velocity of the object is V = (20i + 25j - 12k). The object breaks into two pieces with a mass ratio of 1:5. Let the mass of the smaller piece be m_1 and the mass of the larger piece be m_2. So, m_1 = (1)/(1+5)M = (1)/(6)M. And m_2 = (5)/(1+5)M = (5)/(6)M. The velocity of the smaller piece is v_1 = (100i + 35j + 8k). We need to find the velocity of the larger piece, v_2. Step 2: Apply the principle of conservation of momentum. In the absence of external forces, the total momentum of the system before the explosion is equal to the total momentum after the explosion. P_initial = P_final MV = m_1v_1 + m_2v_2 Step 3: Substitute the known values into the momentum conservation equation. M(20i + 25j - 12k) = (1)/(6)M(100i + 35j + 8k) + (5)/(6)Mv_2 Step 4: Simplify the equation by canceling M from all terms. (20i + 25j - 12k) = (1)/(6)(100i + 35j + 8k) + (5)/(6)v_2 Step 5: Multiply the entire equation by 6 to eliminate fractions. 6(20i + 25j - 12k) = (100i + 35j + 8k) + 5v_2 (120i + 150j - 72k) = (100i + 35j + 8k) + 5v_2 Step 6: Solve for 5v_2. 5v_2 = (120i + 150j - 72k) - (100i + 35j + 8k) 5v_2 = (120 - 100)i + (150 - 35)j + (-72 - 8)k 5v_2 = 20i + 115j - 80k Step 7: Solve for v_2. v_2 = (1)/(5)(20i + 115j - 80k) v_2 = ((20)/(5))i + ((115)/(5))j - ((80)/(5))k v_2 = 4i + 23j - 16k Step 8: Compare the result with the given options. The calculated velocity v_2 = 4i + 23j - 16k matches option (a). The final answer is (a).