# Problem: One of the waste products of a nuclear reactor is plutonium-239 (239Pu). This nucleus is radioactive and decays by splitting into a helium-4 nucleus and a uranium-235 nucleus (4He + 235U), the latter of which is also radioactive and will itself decay some time later. The energy emitted in the plutonium decay is 8.40×10-13 J and is entirely converted to kinetic energy of the helium and uranium nuclei. The mass of the helium nucleus is 6.68×107 kg, while that of the uranium is 3.92×10-25 kg (note that the ratio of the masses is 4 to 235). (a) Calculate the velocities of the two nuclei, assuming the plutonium nucleus is originally at rest.(b) How much kinetic energy does each nucleus carry away? Note that the data given here are accurate to three digits only

###### FREE Expert Solution

Conservation of momentum is expressed as:

$\overline{)\begin{array}{rcl}{\mathbf{p}}_{\mathbf{i}}& {\mathbf{=}}& {\mathbf{p}}_{\mathbf{f}}\\ \mathbf{m}{\mathbf{v}}_{\mathbf{i}}& {\mathbf{=}}& {\mathbf{m}}_{\mathbf{H}\mathbf{e}}{\mathbf{v}}_{\mathbf{H}\mathbf{e}}\mathbf{+}{\mathbf{m}}_{\mathbf{U}}{\mathbf{v}}_{\mathbf{U}}\end{array}}$

(a)

Plutonium is in initially at rest.

So we have:

$\begin{array}{rcl}\mathbf{m}\mathbf{\left(}\mathbf{0}\mathbf{\right)}& \mathbf{=}& \mathbf{\left(}\mathbf{6}\mathbf{.}\mathbf{63}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{\right)}{\mathbf{v}}_{\mathbf{H}\mathbf{e}}\mathbf{+}\mathbf{\left(}\mathbf{3}\mathbf{.}\mathbf{952}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{25}}\mathbf{\right)}{\mathbf{v}}_{\mathbf{U}}\\ \frac{\mathbf{\left(}\mathbf{6}\mathbf{.}\mathbf{63}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{\right)}{\mathbf{v}}_{\mathbf{H}\mathbf{e}}\mathbf{+}\mathbf{\left(}\mathbf{3}\mathbf{.}\mathbf{952}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{25}}\mathbf{\right)}{\mathbf{v}}_{\mathbf{U}}}{\mathbf{6}\mathbf{.}\mathbf{63}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{27}}}& \mathbf{=}& \mathbf{0}\\ \begin{array}{l}\begin{array}{rc}{\mathbf{v}}_{\mathbf{He}}& \mathbf{+}\mathbf{59}\mathbf{.}\mathbf{608}{\mathbf{v}}_{\mathbf{U}}\end{array}\end{array}& \mathbf{=}& \mathbf{0}\end{array}$

vHe = - 59.608vU

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###### Problem Details

One of the waste products of a nuclear reactor is plutonium-239 (239Pu). This nucleus is radioactive and decays by splitting into a helium-4 nucleus and a uranium-235 nucleus (4He + 235U), the latter of which is also radioactive and will itself decay some time later. The energy emitted in the plutonium decay is 8.40×10-13 J and is entirely converted to kinetic energy of the helium and uranium nuclei. The mass of the helium nucleus is 6.68×107 kg, while that of the uranium is 3.92×10-25 kg (note that the ratio of the masses is 4 to 235).
(a) Calculate the velocities of the two nuclei, assuming the plutonium nucleus is originally at rest.
(b) How much kinetic energy does each nucleus carry away? Note that the data given here are accurate to three digits only