To find the kinetic energies of particles A and B, we need to use the principle of conservation of energy. The total energy of the system, which includes both the spring and the particles, remains constant. This means that the energy stored in the spring is transferred entirely to the kinetic energies of particles A and B.
Given that the energy stored in the spring is 132 J, we can assume that this energy is equally divided between particles A and B since they fly off in opposite directions.
Let's denote the kinetic energy of particle A as KA and the kinetic energy of particle B as KB.
Since particle A has a mass 3.00 times that of particle B, we can say:
KA = 3KB
The total energy of the system is the sum of the energies of particles A and B:
132 J = KA + KB
Substituting the value of KA in terms of KB, we get:
132 J = 3KB + KB
Combining like terms:
132 J = 4KB
Now, we can solve for KB:
KB = 132 J / 4
KB = 33 J
Since KA = 3KB, we can calculate KA:
KA = 3 * 33 J
KA = 99 J
Therefore, the kinetic energy of particle A is 99 J, and the kinetic energy of particle B is 33 J.