Question

the symbols below represent two planets.
5 represents a planet with a mass 5 times earths mass.
9 represents a planet with a mass 9 times earths mass
which combination of planet masses and distances produces the greatest gravitational force between the planets?

Explanation:

Step1: Recall the formula for gravitational force

The formula for gravitational force is \(F = G\frac{m_1m_2}{r^2}\), where \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the two objects, and \(r\) is the distance between them.

Step2: Calculate the value of \(\frac{m_1m_2}{r^2}\) for each option

• For option 1: \(m_1 = 5\), \(m_2 = 5\), \(r=100\times10^{6}\). Then \(\frac{m_1m_2}{r^2}=\frac{5\times5}{(100\times 10^{6})^2}=\frac{25}{10^{16}}\)
• For option 2: \(m_1 = 5\), \(m_2 = 5\), \(r = 200\times10^{6}\). Then \(\frac{m_1m_2}{r^2}=\frac{5\times5}{(200\times 10^{6})^2}=\frac{25}{4\times10^{16}}\)
• For option 3: \(m_1 = 9\), \(m_2 = 9\), \(r=100\times10^{6}\). Then \(\frac{m_1m_2}{r^2}=\frac{9\times9}{(100\times 10^{6})^2}=\frac{81}{10^{16}}\)
• For option 4: \(m_1 = 9\), \(m_2 = 9\), \(r = 200\times10^{6}\). Then \(\frac{m_1m_2}{r^2}=\frac{9\times9}{(200\times 10^{6})^2}=\frac{81}{4\times10^{16}}\)

Since \(\frac{81}{10^{16}}>\frac{25}{10^{16}}>\frac{81}{4\times10^{16}}>\frac{25}{4\times10^{16}}\)

Answer:

Option 3