To solve this problem, we will use the inverse square law for gravitational force. This law states that the force of gravity is inversely proportional to the square of the distance between two objects. In this context, the force of gravity (which is your weight) on the surface of the Earth and at some distance away from the center of the Earth can be related using the formula:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
Here, [tex]\( F \)[/tex] is the gravitational force, [tex]\( G \)[/tex] is the gravitational constant, [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses involved, and [tex]\( r \)[/tex] is the distance between the centers of the two masses.
We can write the relationship between your weight at the surface of the Earth and at some other distance from the center of the Earth as:
[tex]\[ \frac{F_1}{F_2} = \left(\frac{R_2}{R_1}\right)^2 \][/tex]
Where:
- [tex]\( F_1 \)[/tex] is your initial weight on the surface of the Earth (600 N),
- [tex]\( F_2 \)[/tex] is the weight you want to achieve (300 N),
- [tex]\( R_1 \)[/tex] is the radius of the Earth ([tex]\(4.4 \times 10^6\)[/tex] meters),
- [tex]\( R_2 \)[/tex] is the new distance from the center of the Earth where your weight will be 300 N.
Firstly, we rearrange the above equation to solve for [tex]\( R_2 \)[/tex]:
[tex]\[ \left( \frac{R_2}{R_1} \right)^2 = \frac{F_1}{F_2} \][/tex]
Taking the square root of both sides:
[tex]\[ \frac{R_2}{R_1} = \sqrt{ \frac{F_1}{F_2}} \][/tex]
[tex]\[ R_2 = R_1 \times \sqrt{\frac{F_1}{F_2}} \][/tex]
Now, substitute the known values into the equation:
[tex]\[ R_2 = 4.4 \times 10^6 \times \sqrt{\frac{600}{300}} \][/tex]
[tex]\[ R_2 = 4.4 \times 10^6 \times \sqrt{2} \][/tex]
Given that [tex]\( \sqrt{2} \approx 1.414 \)[/tex]:
[tex]\[ R_2 \approx 4.4 \times 10^6 \times 1.414 \][/tex]
[tex]\[ R_2 \approx 6222539.674441619 \][/tex] meters
Converting this to scientific notation, we get:
[tex]\[ R_2 \approx 6.23 \times 10^6 \, \textup{meters} \][/tex]
So, to answer the question, you should go to a distance of approximately [tex]\( 6.23 \times 10^6 \)[/tex] meters from the center of the Earth in order for your weight to be 300 N.
