Suppose a supernova explosion throws off \(5 M_{\odot}\) of material at an initial speed of \(10^{3} \mathrm{km} / \mathrm{s} .\) (a) Calculate the initial kinetic energy of the shell and the sum of the magnitudes of the momenta of all the pieces in the shell. (b) Suppose the shell slows down by conservation of momentum in sweeping up interstellar material. How much mass will be swept up before the shell slows to \(10 \mathrm{km} / \mathrm{s} ?(\mathrm{c})\) If the average density of interstellar material is \(1 \mathrm{H}\) atom \(/ \mathrm{cm}^{3},\) what is the radius of the shell when it reaches \(10 \mathrm{km} / \mathrm{s} ?\)

The kinetic energy of a supernova explosion is the energy associated with the motion of the material ejected during the explosion. This energy can be calculated using the equation for kinetic energy: \( E_k = \frac{1}{2}mv^2 \). Here, \( m \) is the mass of the ejected material and \( v \) is its speed.
When a supernova throws off material with a mass of 5 solar masses (\(5 M_{\odot} \)), and an initial speed of 1000 km/s (\(10^3 \) km/s), we can substitute these values into the equation:
\[ m = 5 \cdot 2 \cdot 10^{30} \, kg, \] since a solar mass is approximately \( 2 \cdot 10^{30} \, kg \)
\[ v = 10^3 \, km/s = 10^6 \, m/s. \]
The initial kinetic energy is then: \[ E_k = \frac{1}{2} \cdot 5 \cdot 2 \cdot 10^{30} \cdot (10^6)^2 \, J. \] This calculation provides us the immense amount of energy released in a supernova explosion.

Momentum conservation is a key principle in physics, which states that the total momentum of a closed system remains constant if no external forces act on it. For a supernova explosion, the initial momentum of the shell can be calculated using the formula: \( p = mv \)
For the mass \( m \) of 5 solar masses and an initial speed \( v \) of 1000 km/s, the initial momentum is: \[ p_i = m \cdot v = 5 \times 2 \times 10^{30} \cdot 10^6 \; kg \cdot m/s. \] When the ejected material interacts with the interstellar medium and slows down to 10 km/s, the conservation of momentum can be used to find how much interstellar mass (\( m_{int} \)) it sweeps up:
\[ m_i \cdot v_i = (m_i + m_{int}) \cdot v_f \] Here, \( m_i \) is the initial mass, \( v_i \) is initial speed, \( v_f \) is final speed after slowing to 10 km/s. Solving for \( m_{int} \) lets us determine the swept-up mass.

The density of the interstellar material is an important factor in understanding the dynamics of a supernova shell. The average density of interstellar material is typically given in terms of hydrogen atoms per cubic centimeter, often denoted as \( 1 \; H \; atom/cm^3 \).
This density can help us calculate the volume of space the supernova shell will interact with as it moves through the interstellar medium.
If the density \( \rho \) is \( 1 \; H \; atom/cm^3 \), it can be converted to mass density using the mass of a hydrogen atom (\(\approx 1.67 \cdot 10^{-24} \; g \) or \( 1.67 \cdot 10^{-27} \; kg \)).
The volume \( V \) through which the shell moves can be found using the swept-up mass \( m_{int} \) and the density \( \rho \):
\ applications, understanding the overall density and mass interaction provides insight into how the supernova shell expands and slows down.

The dynamics of a supernova shell describe how the shell of ejected material moves and changes over time as it expands into the interstellar medium. As the shell travels outward, it interacts with the surrounding interstellar material, sweeping it up and conserving momentum. This interaction causes the shell to slow down.
The radius of the shell when it slows to a certain speed, like 10 km/s, can be calculated once we know the swept-up mass and the density of the interstellar material. If the mass density is known, the volume \( V \) can be found using the mass and density relation: \[ m = \rho \cdot V. \]
Using the volume of a sphere: \[ V = \frac{4}{3} \pi R^3 \]
Where \( R \) is the radius of the shell, we can solve for \( R \) based on the swept mass and the density. Understanding these dynamics helps us predict the behavior and the eventual fate of the supernova remnants in space.