Final answer:
The ratio of kinetic energies in nuclear decay can be obtained by first applying the conservation of momentum to find the individual velocities of the helium (alpha particle) and uranium nuclei and then using the kinetic energy formula to calculate the energies. The ratio is the kinetic energy of the alpha particle divided by the kinetic energy of the uranium nucleus.
Step-by-step explanation:
Calculating Ratios of Kinetic Energies in Nuclear Decay
To calculate the velocities of the helium (alpha particle) and uranium nuclei after nuclear decay, one would apply the principle of conservation of momentum. This principle states that if the plutonium nucleus is initially at rest, the momentum of the decay products (helium and uranium) must sum to zero. The formula given is:
p_{alpha} = -p_{uranium}
Where m_{alpha} is the mass of the alpha particle, v_{alpha} is the velocity of the alpha particle, m_{uranium} is the mass of the uranium nucleus, and v_{uranium} is the velocity of the uranium nucleus. By rearranging the equation, we can find the individual velocities, given that we know the masses of the alpha particle and the uranium nucleus.
To calculate the kinetic energy (KE) carried by each nucleus, we can use the kinetic energy formula:
KE = ½ m v²
Given that the total energy released in the decay is known, we can express the kinetic energies of both alpha particle and uranium nucleus in terms of this total energy, taking into account their mass ratio and velocities. The ratio of kinetic energies can then be calculated as:
KE_{ratio} = KE_{alpha} / KE_{uranium}
Finally, one could calculate the kinetic energy for each particle and then divide the kinetic energy of the alpha particle by the kinetic energy of the uranium nucleus to obtain the desired ratio.