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A radioactive nucleus of mass M emits a photon of frequency $\nu$ and the nucleus recoils. The recoil energy will be
A
$h\nu$
B
$Mc^2 - h\nu$
C
$\frac{h^2\nu^2}{2Mc^2}$
D
Zero
Detailed Solution
Momentum of the emitted photon: $p = \frac{E}{c} = \frac{h\nu}{c}$
The nucleus is initially at rest, so by conservation of linear momentum the nucleus recoils with an equal and opposite momentum: $p_{nucleus} = \frac{h\nu}{c}$
Recoil (kinetic) energy of the nucleus: $E = \frac{p^2}{2M}$
$E = \frac{\left(\frac{h\nu}{c}\right)^2}{2M}$
$E = \frac{h^2\nu^2}{2Mc^2}$
The nucleus is initially at rest, so by conservation of linear momentum the nucleus recoils with an equal and opposite momentum: $p_{nucleus} = \frac{h\nu}{c}$
Recoil (kinetic) energy of the nucleus: $E = \frac{p^2}{2M}$
$E = \frac{\left(\frac{h\nu}{c}\right)^2}{2M}$
$E = \frac{h^2\nu^2}{2Mc^2}$