JEE MAIN - Physics (2010 - No. 3)
A nucleus of mass $$M+$$$$\Delta m$$ is at rest and decays into two daughter nuclei of equal mass $${M \over 2}$$ each. Speed of light is $$c.$$
The speed of daughter nuclei is
$$c{{\Delta m} \over {M + \Delta m}}$$
$$c\sqrt {{{2\Delta m} \over M}} $$
$$c\sqrt {{{\Delta m} \over M}} $$
$$c\sqrt {{{\Delta m} \over {M + \Delta m}}} $$
Explanation
By conservation of energy,
$$\left( {M + \Delta m} \right){c^2} = {{2M} \over 2}{c^2} + {1 \over 2}.{{2M} \over 2}{v^2},$$
where $$v$$ is the speed of the daughter nuclei
$$ \Rightarrow \Delta m{c^2} = {M \over 2}{v^2}$$
$$\therefore$$ $$v = c\sqrt {{{2\Delta m} \over M}} $$
$$\left( {M + \Delta m} \right){c^2} = {{2M} \over 2}{c^2} + {1 \over 2}.{{2M} \over 2}{v^2},$$
where $$v$$ is the speed of the daughter nuclei
$$ \Rightarrow \Delta m{c^2} = {M \over 2}{v^2}$$
$$\therefore$$ $$v = c\sqrt {{{2\Delta m} \over M}} $$
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