Assume that the system described by the differential equation mu+γ u+ku = 0 is critically damped and that the initial conditions are u(0) = u0, u(0) = v0. If v0 = 0, show that u → 0 as t → [infinity] but that u is never zero. If u0 is positive, determine a condition on v0 that will ensure that the mass passes through its equilibrium position after it is released.

V(o) < -[γu(o)]/2m

Explanation:

I've attached the explanation.

answer: the correct answer is 1.

explanation: there are various process in which a radioactive nuclei decays:

1) alpha decay: in this process, a heavier nuclei decays into lighter nuclei by releasing alpha particle. the mass number is reduced by 4 units.

2) electron capture: in this decay process, a parent nuclei absorbs an electron and gets converted into a neutron. simply, a proton and an electron combines together to form a neutron. mass number does not change in this process.

3) gamma ray emission: it is a decay process in which an unstable nuclei gives excess energy by a spontaneous electromagnetic process. this decay releases . this process does not change the mass number.

4) positron emission: it is a type of decay process, in which a proton gets converted to neutron and an electron neutrino. this is also known as -decay. in this the mass number remains same.

from the above information, we see that only alpha-decay is producing a change in the mass number of the nucleus.