Question

There is a stream of neutrons with a kinetic energy of $0.0327eV$. If the half-life of neutron is $700s$, if $\frac{x}{1000}$ fraction of neutrons will decay before they travel a distance $3$ of $10km$. Find $x$.

Answer 1

$KE=\frac{1}{2}m{v}^2=0.0327eV=0.0327\times 1.6\times {10}^{-19}J$

$v^2=\frac{2\times 0.0327\times 1.6\times {10}^{-19}}{1.008\times 1.66\times {10}^{-27}}$

$v=2.5\times {10}^{3}m/s$
Time, $t=\frac{S}{v}=\frac{10\times 1000}{2.5\times {10}^3}=4s$

Decay constant, $\lambda =\frac{0.693}{t_{1/2}}=\frac{0.693}{700}=9.9\times {10}^{-4}{s}^{-1}$
Now, $\frac{dN}{dt}=-N\lambda$

$\left|\frac{dN}{N}\right|=\lambda dt=9.9\times {10}^{-4}\times (4-0)=0.00396=0.004$