A piece of space debris is travelling in an elliptical orbit around a planet. At its closest point to the planet, the debris is travelling at 5000ms. When the debris is 10000km from the planet, it travels at 5000kmhr. How close does the debris get to the planet in its orbit?

According to Kepler's Second Law of Planetary motion (also known as the law of areas), the angular momentum [tex]L[/tex] is constant, that is, it is preserved.

Therefore we will use the principle of conservation of angular momentum to solve this problem:

Knowing [tex]L_{1}=L_{2}[/tex] (1) Where [tex]L[/tex] is expressed in [tex]kg\frac{m^{2}}{s}[/tex]

Being:

[tex]L_{1}=mv_{1}r_{1}[/tex] (2) and [tex]L_{2}=mv_{2}r_{2}[/tex] (3)

Where:

[tex]m[/tex] is the mass of space debris

[tex]v_{1}=5000m/s[/tex] is the velocity of space debris at apogee (closest point to the Earth)

[tex]r_{1}[/tex] the apogee

[tex]v_{2}=5000km/h=1388.88m/s[/tex] is the velocity of space debris at a distance of 10000km

[tex]r_{2}=10000km=10000000m[/tex]

Substituting (2) and (3) in (1):

[tex]mv_{1}r_{1}=mv_{2}r_{2}[/tex] (4)

Isolating [tex]r_{1}[/tex] and solving:

[tex]r_{1}=\frac{v_{2}r_{2}}{v_{1}}[/tex] (5)

[tex]r_{1}=\frac{(1388.88m/s)(10000000m)}{5000m/s}[/tex] (6)

[tex]r_{1}=2777760m=2777.76km[/tex] (7)

Finally:

[tex]r_{1}=2777.76km \approx 2778km [/tex]