To solve these two scenarios involving the determination of tangential speeds, we need to decide the proper formula to use based on the given information in each case. Here are the detailed steps:
### Scenario 1:
Determine the tangential speed of the moon given the mass of Earth and the distance from Earth to the moon.
- Relevant information:
- Mass of Earth ([tex]\(m_{\text{central}}\)[/tex]) is given.
- Distance (radius [tex]\(r\)[/tex]) from Earth to the moon is given.
- Formula considerations:
- Formula A: [tex]\( v = \frac{2 \pi r}{T} \)[/tex], where [tex]\(T\)[/tex] is the orbital period.
- Formula B: [tex]\( v^2 = G \frac{m_{\text{central}}}{r} \)[/tex], where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( m_{\text{central}} \)[/tex] is the mass of the central object (in this case, Earth), and [tex]\( r \)[/tex] is the radius (distance from the Earth to the moon).
Since we are given the mass of the central object (Earth) and the radius (distance from Earth to the moon), but not the orbital period [tex]\(T\)[/tex], we should choose formula B for this scenario.
Answer for Scenario 1: [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.
- Relevant information:
- Orbital period ([tex]\(T\)[/tex]) is given as 90 minutes.
- Distance (radius [tex]\(r\)[/tex]) above Earth's surface is given, which is 150 km. Note: this distance should be added to Earth's radius for accurate calculations, but that's not necessary for this step.
- Formula considerations:
- Formula A: [tex]\( v = \frac{2 \pi r}{T} \)[/tex], which requires the orbital period [tex]\(T\)[/tex] and radius [tex]\(r\)[/tex].
- Formula B: [tex]\( v^2 = G \frac{m_{\text{central}}}{r} \)[/tex], which requires the mass of the central object and the distance.
Since we are given the orbital period ([tex]\(T\)[/tex]) and the radius ([tex]\(r\)[/tex]), we should use formula A for this scenario.
Answer for Scenario 2: [tex]\( \boxed{A} \)[/tex]
### Summary:
- For the first scenario, we use formula B.
- For the second scenario, we use formula A.
Therefore, the final answers are:
- Scenario 1: [tex]\( \boxed{B} \)[/tex]
- Scenario 2: [tex]\( \boxed{A} \)[/tex]
