Answer :
Let's determine which formula to use for each scenario by analyzing the given information step by step.
### Scenario 1:
Determine the tangential speed of the moon given the mass of Earth and the distance from Earth to the moon.
To solve this scenario, we need to consider the formula that involves the given mass of the central object (Earth) and the distance to the orbiting object (moon).
The appropriate formula that involves these variables is:
[tex]\[ v^2 = G \frac{m_{\text{central}}}{r} \][/tex]
This formula relates the tangential speed (v) to the gravitational constant (G), the mass of the central object (m_{\text{central}} in this case, the Earth), and the distance from the central object (r, the distance to the moon).
Thus, for Scenario 1, the appropriate formula to use is:
[tex]\[ \boxed{B} \][/tex]
### Scenario 2:
Determine the tangential speed of a satellite that takes 90 minutes to complete an orbit 150 km above Earth's surface.
In this scenario, the time period of the orbit (T) and the orbital radius (r) are given. To find the tangential speed, we can use the formula that directly relates these variables:
[tex]\[ v = \frac{2 \pi r}{T} \][/tex]
This formula allows us to calculate the tangential speed (v) using the radius of the orbit (r) and the orbital period (T). Here, [tex]\( r \)[/tex] would be the sum of Earth's radius and the 150 km altitude, and [tex]\( T \)[/tex] is the given time period of 90 minutes.
Thus, for Scenario 2, the appropriate formula to use is:
[tex]\[ \boxed{A} \][/tex]
### Summary:
- Scenario 1 (tangential speed of the moon): [tex]\( \boxed{B} \)[/tex]
- Scenario 2 (tangential speed of a satellite): [tex]\( \boxed{A} \)[/tex]
### Scenario 1:
Determine the tangential speed of the moon given the mass of Earth and the distance from Earth to the moon.
To solve this scenario, we need to consider the formula that involves the given mass of the central object (Earth) and the distance to the orbiting object (moon).
The appropriate formula that involves these variables is:
[tex]\[ v^2 = G \frac{m_{\text{central}}}{r} \][/tex]
This formula relates the tangential speed (v) to the gravitational constant (G), the mass of the central object (m_{\text{central}} in this case, the Earth), and the distance from the central object (r, the distance to the moon).
Thus, for Scenario 1, the appropriate formula to use is:
[tex]\[ \boxed{B} \][/tex]
### Scenario 2:
Determine the tangential speed of a satellite that takes 90 minutes to complete an orbit 150 km above Earth's surface.
In this scenario, the time period of the orbit (T) and the orbital radius (r) are given. To find the tangential speed, we can use the formula that directly relates these variables:
[tex]\[ v = \frac{2 \pi r}{T} \][/tex]
This formula allows us to calculate the tangential speed (v) using the radius of the orbit (r) and the orbital period (T). Here, [tex]\( r \)[/tex] would be the sum of Earth's radius and the 150 km altitude, and [tex]\( T \)[/tex] is the given time period of 90 minutes.
Thus, for Scenario 2, the appropriate formula to use is:
[tex]\[ \boxed{A} \][/tex]
### Summary:
- Scenario 1 (tangential speed of the moon): [tex]\( \boxed{B} \)[/tex]
- Scenario 2 (tangential speed of a satellite): [tex]\( \boxed{A} \)[/tex]
