Final answer:
The ratio of the masses m₁ and m₂ is determined to be 4:1 by using the conservation of momentum and kinetic energy principles during an explosion.
Step-by-step explanation:
The student's question pertains to the conservation of momentum and kinetic energy during an explosion where two fragments are generated. In such a scenario, the total momentum should be conserved, meaning the momentum of each fragment should be equal and opposite because the initial momentum was zero.
In terms of kinetic energy, given that the first fragment (with mass m1) acquires twice the kinetic energy of the second one (with mass m2), the kinetic energy can be represented as KE1 = 2KE2. Since kinetic energy for each fragment is given by KE = (1/2)mv2, we can set up a ratio as such: (1/2)m1v12 = 2(1/2)m2v22, which simplifies to m1v12 = 2m2v22.
From the conservation of momentum, m1v1 = m2v2, we can substitute v2 = (m1/m2)v1 into the energy equation to ultimately find the ratio of the masses. This yields m1v12 = 2m2((m1/m2)v1)2, which simplifies to m1 = 4m2. Hence, the ratio of their masses m1/m2 is 4:1.