Question

The Enterprise wants to orbit a4.15 x 10^24kg planet with a period of14100 s. What should the radius oftheir orbit be?

Answer 1

Given:

Mass of planet, m = 4.15 x 10²⁴ kg.

Period, T = 14100 s

Let's find the radius.

To find the radius, apply the formula from Kepler's Third aw:

$\begin{gathered} \frac{T^2}{R^3}=\frac{4\pi^2}{GM} \\ \\ \end{gathered}$

Where R is the radius.

Rewrite the formula for r:

$R=\sqrt[3]{\frac{GM*T^2}{4\pi^2}}$

Where:

G is gravitational constant = 6.67 x 10⁻¹¹ m3 kg-1 s-2

M is the mass = 4.15 x 10²⁴ kg

T is the period = 14100 s

π = 3.54

Plug in values and solve for R;

$\begin{gathered} R=\sqrt[3]{\frac{6.67\times10^{-11}*4.15\times10^{24}*14100^2}{4\pi^2}} \\ \\ R=\sqrt[3]{\frac{5.503\times10^{22}}{39.4784}} \\ \\ \end{gathered}$

Solving further:

$\begin{gathered} R=\sqrt[3]{1.394\times10^{21}} \\ \\ R=11170796.49\approx1.12\times10^7\text{ m} \end{gathered}$

Therefore, the orbital radius will be 1.12 x 10⁷ meters.

ANSWER:

1.12 x 10⁷ m