The orbital period of the spacecraft is about 5.9 × 10³ s ≈ 99 minutes
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<h3>Further explanation</h3>
<em>Let's recall </em><em>Centripetal Force </em><em>formula as follows:</em>
![\boxed{F = m \frac{v^2}{R}}](https://tex.z-dn.net/?f=%5Cboxed%7BF%20%3D%20m%20%5Cfrac%7Bv%5E2%7D%7BR%7D%7D)
<em>where:</em>
<em>F = Centripetal Force ( Newton )</em>
<em>m = mass of object ( kg )</em>
<em>v = speed of object ( m/s )</em>
<em>R = radius ( m )</em>
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<em>Newton's gravitational law </em><em>states that the force of attraction between two objects can be formulated as follows:</em>
![\boxed {F = G \frac{m_1 ~ m_2}{R^2}}](https://tex.z-dn.net/?f=%5Cboxed%20%7BF%20%3D%20G%20%5Cfrac%7Bm_1%20~%20m_2%7D%7BR%5E2%7D%7D)
<em>where:</em>
<em>F = Gravitational Force ( Newton )</em>
<em>G = Gravitational Constant ( 6.67 × 10⁻¹¹ Nm² / kg² )</em>
<em>m = Object's Mass ( kg )</em>
<em>R = Distance Between Objects ( m )</em>
Let us now tackle the problem !
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<u>Given:</u>
free-fall acceleration = g = 3.8 m/s²
radius of Mars = R = 3.37 × 10⁶ m
<u>Asked:</u>
orbital period = T = ?
<u>Solution:</u>
![\Sigma F = ma](https://tex.z-dn.net/?f=%5CSigma%20F%20%3D%20ma)
![G \frac{ M m} { R^2 } = m \omega^2 R](https://tex.z-dn.net/?f=G%20%5Cfrac%7B%20M%20m%7D%20%7B%20R%5E2%20%7D%20%3D%20m%20%5Comega%5E2%20R)
![G \frac{ M } { R^2 } = \omega^2 R](https://tex.z-dn.net/?f=G%20%5Cfrac%7B%20M%20%7D%20%7B%20R%5E2%20%7D%20%3D%20%5Comega%5E2%20R)
![g = \omega^2 R](https://tex.z-dn.net/?f=g%20%3D%20%5Comega%5E2%20R)
![\omega^2 = g \div R](https://tex.z-dn.net/?f=%5Comega%5E2%20%3D%20g%20%5Cdiv%20R)
![\omega = \sqrt { g \div R }](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Csqrt%20%7B%20g%20%5Cdiv%20R%20%7D)
![2 \pi \div T = \sqrt { g \div R }](https://tex.z-dn.net/?f=2%20%5Cpi%20%5Cdiv%20T%20%3D%20%5Csqrt%20%7B%20g%20%5Cdiv%20R%20%7D)
![T = 2 \pi \div \sqrt { g \div R](https://tex.z-dn.net/?f=T%20%3D%202%20%5Cpi%20%5Cdiv%20%5Csqrt%20%7B%20g%20%5Cdiv%20R%20)
![T = 2 \pi \sqrt{ R \div g}](https://tex.z-dn.net/?f=T%20%3D%202%20%5Cpi%20%5Csqrt%7B%20R%20%5Cdiv%20g%7D)
![T = 2 \pi \sqrt{ 3.37 \times 10^6 \div 3.8 }](https://tex.z-dn.net/?f=T%20%3D%202%20%5Cpi%20%5Csqrt%7B%203.37%20%5Ctimes%2010%5E6%20%5Cdiv%203.8%20%7D)
![\boxed{T \approx 5.9 \times 10^3 \texttt{ s}}](https://tex.z-dn.net/?f=%5Cboxed%7BT%20%5Capprox%205.9%20%5Ctimes%2010%5E3%20%5Ctexttt%7B%20s%7D%7D)
![\boxed{T \approx 99 \texttt{ minutes}}](https://tex.z-dn.net/?f=%5Cboxed%7BT%20%5Capprox%2099%20%5Ctexttt%7B%20minutes%7D%7D)
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<h3>Learn more</h3>
![\texttt{ }](https://tex.z-dn.net/?f=%5Ctexttt%7B%20%7D)
<h3>Answer details</h3>
Grade: High School
Subject: Physics
Chapter: Gravitational Fields