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.
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.
