Explosion of a probe

Problem

A probe of mass $m = 10^4\ \text{kg}$ orbits the Earth at distance $R$. It explodes into two equal fragments ($m_1 = m_2 = 5\cdot 10^3\ \text{kg}$), releasing an energy $E_\text{expl} = 2{,}24\cdot 10^{32}\ \text{J}$. The two fragments fly apart horizontally in opposite directions with the same speed relative to the centre of mass. Imposing conservation of momentum and energy, find the speeds $v_{1C}$ and $v_{2C}$ of the two fragments relative to the CM.

Solution

Conservation of momentum. In the centre-of-mass frame the total momentum is zero before and after the explosion: $m_1 v_{1C} - m_2 v_{2C} = 0$

With $m_1 = m_2$ it follows immediately that $\ev{v_{1C} = v_{2C} \equiv v}$ i.e. the two fragments have the same speed (in magnitude) and opposite directions.

Conservation of energy. The energy released by the explosion becomes kinetic energy of the two fragments: $E_\text{expl} = \tfrac{1}{2}m_1 v_{1C}^2 + \tfrac{1}{2}m_2 v_{2C}^2 = 2\cdot\tfrac{1}{2}m_\text{fr}\,v^2 = m_\text{fr}\,v^2$

Solving for $v$ (with $m_\text{fr}=5\cdot 10^3\ \text{kg}$, equivalent to $v=\sqrt{2E_\text{expl}/m}$ using the total mass): $v = \sqrt{\frac{E_\text{expl}}{m_\text{fr}}} = \sqrt{\frac{2{,}24\cdot 10^{32}}{5\cdot 10^{3}}} = \sqrt{4{,}48\cdot 10^{28}}$

$\ev{v_{1C} = v_{2C} \approx 2{,}1\cdot 10^{14}\ \text{m/s}}$

Watch the order of magnitude (science fiction alert). The result is enormously greater than the speed of light ($c\approx 3\cdot 10^8\ \text{m/s}$): physically impossible. The data of the problem are unrealistic — the assumed energy $E_\text{expl}$ is wildly disproportionate for masses of a few thousand kg. The method (momentum + energy) is correct; it is the order of magnitude of the data that must be checked. The textbook answer key gives $\approx 6{,}7\cdot 10^{13}\ \text{m/s}$, consistent with a value of $E_\text{expl}$ ten times smaller ($2{,}24\cdot 10^{31}\ \text{J}$): either way, the result remains unphysical.

Links

Topics: Gravitation Concepts: Conservation of momentum · Explosion · Kinetic energy Skills: Conservation of momentum Objects: Satellite