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Answer :
To find the mass of the astronaut, we need to understand the relationship between weight, mass, and gravity. On a celestial body like the moon, weight is the force due to gravity acting on an object's mass.
The weight [tex]\( F \)[/tex] of an object is calculated using the formula:
[tex]\[ F = G \times \frac{m \times M}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the weight (in Newtons),
- [tex]\( G \)[/tex] is the universal gravitational constant [tex]\( (6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2) \)[/tex],
- [tex]\( m \)[/tex] is the mass of the object (what we're solving for, in kg),
- [tex]\( M \)[/tex] is the mass of the celestial body (moon, in this case),
- [tex]\( r \)[/tex] is the radius of the celestial body.
Rearrange the formula to solve for the mass of the astronaut ([tex]\( m \)[/tex]):
[tex]\[ m = \frac{F \times r^2}{G \times M} \][/tex]
Given values:
- [tex]\( F = 300 \)[/tex] N
- [tex]\( M = 7.35 \times 10^{22} \)[/tex] kg
- [tex]\( r = 1.74 \times 10^6 \)[/tex] meters
- [tex]\( G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2 \)[/tex]
Substitute these values into the equation:
[tex]\[ m = \frac{300 \times (1.74 \times 10^6)^2}{6.674 \times 10^{-11} \times 7.35 \times 10^{22}} \][/tex]
When you calculate this, the result is approximately 185.16 kg. Therefore, the astronaut's mass is closest to:
b) 185 kg
The weight [tex]\( F \)[/tex] of an object is calculated using the formula:
[tex]\[ F = G \times \frac{m \times M}{r^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the weight (in Newtons),
- [tex]\( G \)[/tex] is the universal gravitational constant [tex]\( (6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2) \)[/tex],
- [tex]\( m \)[/tex] is the mass of the object (what we're solving for, in kg),
- [tex]\( M \)[/tex] is the mass of the celestial body (moon, in this case),
- [tex]\( r \)[/tex] is the radius of the celestial body.
Rearrange the formula to solve for the mass of the astronaut ([tex]\( m \)[/tex]):
[tex]\[ m = \frac{F \times r^2}{G \times M} \][/tex]
Given values:
- [tex]\( F = 300 \)[/tex] N
- [tex]\( M = 7.35 \times 10^{22} \)[/tex] kg
- [tex]\( r = 1.74 \times 10^6 \)[/tex] meters
- [tex]\( G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2 \)[/tex]
Substitute these values into the equation:
[tex]\[ m = \frac{300 \times (1.74 \times 10^6)^2}{6.674 \times 10^{-11} \times 7.35 \times 10^{22}} \][/tex]
When you calculate this, the result is approximately 185.16 kg. Therefore, the astronaut's mass is closest to:
b) 185 kg
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