slowing down of neutrons: In a nuclear reactor a neutron of high speed (typically $10^{7}\; m s ^{-1}$ ) must be slowed to $10^{3}\; m s ^{-1}$ so that it can have a high probability of interacting with isotope $^{235} _{92} U$ and causing it to fission. Show that a neutron can lose most of its kinetic energy In an elastic collision with a light nuclel like deuterlum or carbon which has a mass of only a few times the neutron mass. The material making up the light nuclel, usually heavy water $\left( D _{2} O \right)$ or graphite, is called a moderator.
The initial kinetic energy of the neutron is
$K_{1 i}=\frac{1}{2} m_{1} v_{1 i}^{2}$
while its final kinetic energy from Eq. $(6.27)$
$K_{1 f}=\frac{1}{2} m_{1} v_{1 f}^{2}=\frac{1}{2} m_{1}\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} v_{1 t}^{2}$
The fractional kinetic energy lost is
$f_{1}=\frac{K_{1 f}}{K_{1 i}}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2}$
while the fractional kinetic energy gained by the moderating nucle $K_{2 f} / K_{1 i}$ is $f_{2}=1-f_{1}$ (elastic collision)
$=\frac{4 m_{1} m_{2}}{\left(m_{1}+m_{2}\right)^{2}}$
One can also verify this result by substituting from Equation
For deuterium $m_{2}=2 m_{1}$ and we obtain $f_{1}=1 / 9$ while $f_{2}=8 / 9 .$ Almost $90 \%$ of the neutron's energy is transferred to deuterlum. For carbon $f_{1}=71.6 \%$ and $f_{2}=28.4 \% .$ In practice however, this number is smaller since head-on collisions are rare.
A block of mass $M$ is attached to the lower end of a vertical spring. The spring is hung from a ceiling and has force constant value $k.$ The mass is released from rest with the spring initially unstretched. The maximum extension produced in the length of the spring will be
A one kg block moves towards a light spring with a velocity of $8\, m/s$. When the spring is compressed by $3\, m$, its momentum becomes half of the original momentum. Spring constant of the spring is :-
A block $C$ of mass $m$ is moving with velocity $v_0$ and collides elastically with block $A$ of mass $m$ which connected to another block $B$ of mass $2\,m$ through a spring of spring constant $k$. What is $k$ if $x_0$ is the compression of spring when velocity of $A$ and $B$ is same?
A ball of mass $m_1$ falls from height $h_1$ from rest to strike a spring of force constant $K$, which forces another ball of mass $m_2$ to jump on a horizontal floor at a height $h_2$ below from it. Find the horizontal distance at which ball of mass $m_2$ strikes from the position of start :- [Spring does not move]
$A$ block of mass $m$ moving with a velocity $v_0$ on a smooth horizontal surface strikes and compresses a spring of stiffness $k$ till mass comes to rest as shown in the figure. This phenomenon is observed by two observers:
$A$: standing on the horizontal surface
$B$: standing on the block
According to observer $B$, the potential energy of the spring increases