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------------------------------------------------ An astronaut with a mass of 95 kg is on the planet Mercury. What is the weight of the astronaut on Mercury in pounds?

(Note: [tex]$1 \text{ lb} = 4.45 \text{ N}$[/tex])

- Mercury Mass: [tex]$3.3 \times 10^{23} \text{ kg}$[/tex]
- Mercury Radius: [tex]$2,439,500 \text{ m}$[/tex]

Choose the correct weight:

A. 157 lbs
B. 168 lbs
C. 47 lbs
D. 79 lbs
E. 26 lbs
F. 210 lbs
G. 32 lbs
H. 114 lbs

Answer :

Sure, let's solve this step-by-step!

1. Understand the Problem:
We need to find the weight of an astronaut (who has a mass of 95 kg) on the planet Mercury, in pounds (lbs).

2. Use the Correct Formula:
The weight of an object on a planet is calculated using the formula for gravitational force:
[tex]\[
F = \frac{G \times m_1 \times m_2}{r^2}
\][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force (weight) in Newtons,
- [tex]\( G \)[/tex] is the gravitational constant ([tex]\(6.67430 \times 10^{-11} \, m^3 kg^{-1} s^{-2}\)[/tex]),
- [tex]\( m_1 \)[/tex] is the mass of the astronaut (95 kg),
- [tex]\( m_2 \)[/tex] is the mass of Mercury ([tex]\(3.3 \times 10^{23} \, kg\)[/tex]),
- [tex]\( r \)[/tex] is the radius of Mercury (2,439,500 m).

3. Calculate the Gravitational Force:
Plug in the values to find the force in Newtons:

[tex]\[
F = \frac{6.67430 \times 10^{-11} \times 95 \times 3.3 \times 10^{23}}{(2,439,500)^2}
\][/tex]

This calculation results in approximately 351.59 Newtons.

4. Convert to Pounds:
The problem states that 1 pound is equivalent to 4.45 Newtons. So to convert the weight from Newtons to pounds:

[tex]\[
\text{Weight in lbs} = \frac{351.59}{4.45}
\][/tex]

This calculation gives us approximately 79 lbs.

5. Conclude:
The weight of the astronaut on Mercury is approximately 79 lbs.

Therefore, the astronaut would weigh about 79 pounds on Mercury.