Question

There is a stream of neutrons with a kinetic energy of $0.0327\, eV$. If the half life of neutron is $700$ second, what fraction of neutrons will decay before they travel a distance of $10 m$ ? Given mass of neutron $=1.675^{\prime} 10^{-27} kg$.

• A. $3.96^{\prime} 10^{-2}$
• B. $3.96^{\prime} 10^{-3}$
• C. $3.96^{\prime} 10^{-6}$
• D. $3.96^{\prime} 10^{-8}$

Answer 1

Answer: (C)

From the given kinetic energy of the neutrons, we first calculate their velocity. Thus
$\frac{1}{2} m u^{2}=0.0327^{\prime} 1.6^{\prime} 10^{-19}$
$u^{2}=\frac{2^{\prime} 0.0327^{\prime} 1.6^{\prime} 10^{-19}}{1.675^{\prime} 10^{-27}}=625^{\prime} 10^{4} $
or $ u=2500\, m / s$
with this speed, the time taken by the neutrons to travel a distance of $10 m$ is,
$ D t=\frac{10}{2500}=4^{\prime} 10^{-3} s$
The fraction of neutrons decayed in time $D t$ second is, $\frac{ D N}{N}=l D t \&$ also, $l=\frac{0.693}{T_{1 / 2}}$
$ \frac{ D N}{N}=\frac{0.693}{T_{1 / 2}} D t=\frac{0.693}{700} \cdot\left(4^{\prime} 10^{-3}\right)=3.96^{\prime} 10^{-6} $.