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:

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

v_{He}_{ }= - 59.608v_{U}

One of the waste products of a nuclear reactor is plutonium-239 (^{239}Pu). This nucleus is radioactive and decays by splitting into a helium-4 nucleus and a uranium-235 nucleus (^{4}He + ^{235}U), 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 6.68×10^{7} 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

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