Question

One way to attack a satellite in Earth orbit is to launch a swarm of pellets in the same orbit as the satellite but in the opposite direction. Suppose a satellite in a circular orbit $\mathbf{500}\mathbf{km}$above Earth’s surface collides with a pellet having mass $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{g}$.

(a) What is the kinetic energy of the pellet in the reference frame of the satellite just before the collision?

b) What is the ratio of this kinetic energy to the kinetic energy of a $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{g}$bullet from a modern army rifle with a muzzle speed of $\mathbf{950}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$?

Short answer

1. The kinetic energy of the pellet in the reference frame of the satellite just before the collision is $4.6\times {10}^5\mathrm{J}$
2. The ratio of this kinetic energy to the kinetic energy of a $4.0\mathrm{g}$ bullet from a modern army rifle with a muzzle speed of $950\text{\hspace{0.17em}m/s}$ is $2.6\times {10}^2$

Step-by-step solution

Listing the given quantities

Mass of the bullet$4.0\mathrm{g}$

Muzzle speed of$950\text{m/s}$

Satellite in a circular orbit at$500\text{\hspace{0.17em}km}$above the earth's surface

Understanding the concept of the frame of reference and kinetic energy

Using the formula of kinetic energy, wecan find thekinetic energy of the pellet in the reference frame of the satellite just before the collision and the ratio of this kinetic energy to the kinetic energy of a $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{g}$bullet from a modern army rifle with a muzzle speed of$\mathbf{950}\mathbf{\text{\hspace{0.17em}m/s}}$

Formula:

$\mathbf{\text{K}}\mathbf{=}\frac{\mathbf{\text{1}}}{\mathbf{\text{2}}}{\mathbf{\text{mv}}}^{\mathbf{\text{2}}}$

$\mathbf{\text{v=}}\sqrt{\frac{\mathbf{\text{GM}}}{\mathbf{\text{R}}}}$

(a) Calculation of the kinetic energy of the pellet in the reference frame of the satellite just before the collision  

As $\text{v}=\sqrt{\frac{\text{GM}}{\text{R}}}$and pellets have the same speed but in the opposite direction of motion, so the relative speed between the pellet and satellite is ${\text{v}}_{\text{rel}}=\text{2v}$

$\begin{array}{c}{\text{K}}_{\text{rel}}=\frac{\text{1}}{\text{2}}\text{m}{\left(\text{2}\sqrt{\frac{\text{GM}}{\text{R}}}\right)}^{\text{2}}\\ =\frac{\text{2GMm}}{\text{R}}\end{array}$

As$\mathrm{R}=(6370\times {10}^3\mathrm{m}+500\times {10}^3\mathrm{m})=6870\times {10}^3\mathrm{m}$

$\begin{array}{c}{\text{K}}_{\text{rel}}=\frac{2\left(6.67\times {10}^{-11}\frac{{\mathrm{m}}^3}{\mathrm{kg}\cdot {\mathrm{s}}^2}\right)(5.98\times {10}^{24}\mathrm{kg})(0.0040\mathrm{kg})}{6870\times {10}^3\mathrm{m}}\\ =4.6\times {10}^5\mathrm{J}\end{array}$

Therefore, the kinetic energy of the pellet in the reference frame of the satellite just before the collision is $4.6\times {10}^5\mathrm{J}$

(b) Calculation of the ratio of this kinetic energy to the kinetic energy of a 4.0 g bullet from a modern army rifle with a muzzle speed of   950 m/s 

$\text{K}=\frac{\text{1}}{\text{2}}{\text{mv}}^{\text{2}}$

Therefore,

$\begin{array}{c}\frac{{\text{K}}_{\text{rel}}}{{\text{K}}_{\text{bullet}}}=\frac{4.6\times {10}^5\mathrm{J}}{\frac{1}{2}\left(0.0040\mathrm{kg}\right){\left(950\frac{\mathrm{m}}{{\mathrm{s}}^2}\right)}^2}\\ =\frac{4.6\times {10}^5\mathrm{J}}{1805\mathrm{J}}\\ =2.6\times {10}^2\end{array}$

Therefore, theratio of this kinetic energy to the kinetic energy of a $4.0\text{g}$bullet from a modern army rifle with a muzzle speed of$950\text{m/s}$is$2.6\times {10}^2$.