Question

planet x and planet y have masses of \\(4.60 \times 10^{23}\text{ kg}\\) and \\(1.89 \times 10^{24}\text{ kg}\\) respectively. if they are \\(2.33 \times 10^6\text{ m}\\) from each other, what is the force of gravity felt between them?

\\(\bigcirc\\) \\(1.92 \times 10^{24}\text{ n}\\)
\\(\bigcirc\\) \\(1.07 \times 10^{25}\text{ n}\\)
\\(\bigcirc\\) \\(1.07 \times 10^{23}\text{ n}\\)
\\(\bigcirc\\) \\(2.38 \times 10^{25}\text{ n}\\)

Explanation:

Identify the given values and formula

We are given the masses of two planets and the distance between them:

• Mass of Planet X, \(m_1 = 4.60 \times 10^{23}\text{ kg}\)
• Mass of Planet Y, \(m_2 = 1.89 \times 10^{24}\text{ kg}\)
• Distance between them, \(r = 2.33 \times 10^6\text{ m}\)

To find the gravitational force between them, we use Newton's Law of Universal Gravitation:

$$F = G \frac{m_1 m_2}{r^2}$$

where \(G\) is the universal gravitational constant:

$$G \approx 6.674 \times 10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2$$

Substitute the values into the equation

Using Scientific Notation to organize our calculation, we substitute the values:

$$F = (6.674 \times 10^{-11}) \frac{(4.60 \times 10^{23})(1.89 \times 10^{24})}{(2.33 \times 10^6)^2}$$

Calculate the numerator

First, multiply the coefficients in the numerator:

$$6.674 \times 4.60 \times 1.89 \approx 58.024$$

Next, combine the powers of 10 in the numerator:

$$10^{-11} \times 10^{23} \times 10^{24} = 10^{-11 + 23 + 24} = 10^{36}$$

Thus, the numerator is:

$$58.024 \times 10^{36}$$

Calculate the denominator

Square the distance in the denominator:

$$(2.33 \times 10^6)^2 = (2.33)^2 \times (10^6)^2 \approx 5.429 \times 10^{12}$$

Divide to find the final force

Divide the numerator by the denominator:

$$F \approx \frac{58.024 \times 10^{36}}{5.429 \times 10^{12}}$$

Divide the coefficients:

$$\frac{58.024}{5.429} \approx 10.688$$

Subtract the exponents:

$$10^{36 - 12} = 10^{24}$$

Combine the results:

$$F \approx 10.688 \times 10^{24}\text{ N} \approx 1.07 \times 10^{25}\text{ N}$$

Answer:

• O \(1.92 \times 10^{24}\text{ N}\)
• O \(1.07 \times 10^{25}\text{ N}\) (Correct answer)
• O \(1.07 \times 10^{23}\text{ N}\)
• O \(2.38 \times 10^{25}\text{ N}\)