Calculate the gravitational force [tex]\((F)\)[/tex] between the two objects in the following scenario. Be sure to show all of your work. Remember [tex]\(G\)[/tex] is equal to [tex]\(6.67 \times 10^{-11}\)[/tex].

What is the gravitational force between a [tex]\(100 \, \text{kg}\)[/tex] football player and the Earth, which has a mass of [tex]\(5.98 \times 10^{24} \, \text{kg}\)[/tex], with a distance of [tex]\(6.38 \times 10^6 \, \text{m}\)[/tex]?

- Object [tex]\(1 \left(m_1\right): 100 \, \text{kg}\)[/tex] (Football player)
- Object [tex]\(2 \left(m_2\right): 5.98 \times 10^{24} \, \text{kg}\)[/tex] (Earth)
- Distance [tex]\((r): 6.38 \times 10^6 \, \text{meters}\)[/tex]



Answer :

### Step-by-Step Solution:

1. Understanding the Given Variables:
- Gravitational constant, [tex]\( G \)[/tex]: [tex]\( 6.67 \times 10^{-11} \)[/tex] m³ kg⁻¹ s⁻²
- Mass of the football player, [tex]\( m1 \)[/tex]: 100 kg
- Mass of the Earth, [tex]\( m2 \)[/tex]: [tex]\( 5.98 \times 10^{24} \)[/tex] kg
- Distance between the football player and the center of the Earth, [tex]\( r \)[/tex]: [tex]\( 6.38 \times 10^{6} \)[/tex] m

2. Gravitational Force Formula:
The gravitational force [tex]\( F \)[/tex] between two objects is calculated using Newton's law of universal gravitation:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
Where:
- [tex]\( F \)[/tex] is the gravitational force
- [tex]\( G \)[/tex] is the gravitational constant
- [tex]\( m_1 \)[/tex] is the mass of the first object
- [tex]\( m_2 \)[/tex] is the mass of the second object
- [tex]\( r \)[/tex] is the distance between the centers of the two masses

3. Substituting the Values:
Substitute the given values into the formula:
[tex]\[ F = 6.67 \times 10^{-11} \cdot \frac{100 \cdot 5.98 \times 10^{24}}{(6.38 \times 10^6)^2} \][/tex]

4. Calculating the Denominator:
First, calculate the square of the distance, [tex]\( r \)[/tex]:
[tex]\[ r^2 = (6.38 \times 10^6)^2 = 4.0704 \times 10^{13} \, (\text{square meters}) \][/tex]

5. Calculating the Numerator:
Now, compute the product of the masses and [tex]\( G \)[/tex]:
[tex]\[ 100 \cdot 5.98 \times 10^{24} = 5.98 \times 10^{26} \, (\text{kg}^2) \][/tex]
So, the numerator is:
[tex]\[ 6.67 \times 10^{-11} \cdot 5.98 \times 10^{26} = 3.98734 \times 10^{16} \, (\text{N}\cdot\text{m}^2) \][/tex]

6. Final Calculation:
Divide the numerator by the denominator to find the gravitational force [tex]\( F \)[/tex]:
[tex]\[ F = \frac{3.98734 \times 10^{16}}{4.0704 \times 10^{13}} = 979.9088059276147 \, (\text{N}) \][/tex]

### Conclusion:
The gravitational force [tex]\( F \)[/tex] between a 100 kg football player and the Earth, given the distance of [tex]\( 6.38 \times 10^6 \)[/tex] meters, is approximately:
[tex]\[ F \approx 979.91 \, \text{N} \][/tex]
Thus, the gravitational force is approximately 979.91 Newtons.