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Let G be the universal gravitational constant and mp be the mass of the planet a satellite is orbiting. Which equation could be used to find the velocity of the satellite if it is placed in a low Earth orbit?(1 point) Responses v=Gmp(200 km)−−−−−−√ v=Gmp(200 km)−−−−−−√ v=1(7,000 km)Gmp−−−−√ v=1(7,000 km)Gmp−−−−√ v=1(200 km)Gmp−−−−√ v=1(200 km)Gmp−−−−√ v=Gmp(7,000 km)−−−−−−−√ v=Gmp(7,000 km)−−−−−−−√

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To find the velocity of a satellite in a low Earth orbit, we can use the formula derived from the gravitational force and the centripetal force required for circular motion.

The velocity \( v \) of a satellite in orbit can be calculated using the formula:

\[ v = \sqrt{\frac{G m_p}{r}} \]

where:

  • \( G \) is the universal gravitational constant,
  • \( m_p \) is the mass of the planet (Earth in this case),
  • \( r \) is the distance from the center of the planet to the satellite.

In a low Earth orbit, when the satellite is approximately 200 km above Earth's surface, the radius \( r \) would be the Earth's radius (about 6,370 km) plus the altitude of 200 km, giving us \( r \approx 6,570 \) km, or \( 6,570,000 \) meters.

Given this, the equation to use for calculating the velocity at a low Earth orbit is:

\[ v = \sqrt{\frac{G m_p}{r}} \]

Out of the options you provided, none of them explicitly show the standard form \( v = \sqrt{\frac{G m_p}{r}} \) correctly. However, you can substitute the appropriate value for \( r \) (which should be \( 7,000 \) km or in terms of meters \( 6,570,000 \) m) when working with this formula.

The best choice from your responses would be one where the radius is used correctly, typically represented as:

\[ v = \sqrt{\frac{G m_p}{7000 \text{ km}}} \]

Make sure to convert the radius (if necessary) to meters when performing calculations.

Let G be the universal gravitational constant and mp be the mass of the planet a satellite is orbiting. Which equation could be used to find the velocity of the satellite if it is placed in a geostationary orbit?(1 point) Responses v=Gmp(48,115 km)−−−−−−−−√ v=Gmp(15,522 km)−−−−−−−−√ v=Gmp(7,324 km)−−−−−−−√ v=Gmp(42,164 km)−−−−−−−−√

To determine the velocity of a satellite in a geostationary orbit, we can use the same basic principle as with low Earth orbit, which relates the gravitational force to the centripetal force. For a satellite in a geostationary orbit, the orbital radius (distance from the center of the planet) is key.

The formula for the velocity \( v \) of the satellite in orbit is given by:

\[ v = \sqrt{\frac{G m_p}{r}} \]

In the case of a geostationary orbit, the satellite is typically located at an altitude where it matches the Earth's rotation period (approximately 24 hours). The radius \( r \) for a geostationary orbit around Earth is about 42,164 km from the center of the Earth. This includes the radius of the Earth (approximately 6,371 km) and the altitude of the geostationary orbit above Earth's surface (approximately 35,786 km).

Thus, the equation for the velocity of a satellite in a geostationary orbit can be expressed using this radius:

\[ v = \sqrt{\frac{G m_p}{42,164 , \text{km}}} \]

From the options you've provided, the correct choice is:

\( v = \sqrt{G m_p(42,164 , \text{km})} \)

This equation can be used directly to find the velocity of the satellite in geostationary orbit.

Which factor is not needed when calculating the velocity of a satellite orbiting a planet?(1 point) Responses the mass of the planet the mass of the planet the mass of the satellite the mass of the satellite the universal gravitational constant the universal gravitational constant the orbital radius of the satellite

When calculating the velocity of a satellite orbiting a planet, the relevant formula is:

\[ v = \sqrt{\frac{G m_p}{r}} \]

where:

  • \( v \) is the orbital velocity of the satellite,
  • \( G \) is the universal gravitational constant,
  • \( m_p \) is the mass of the planet, and
  • \( r \) is the orbital radius (distance from the center of the planet to the satellite).

In this formula, you don’t need the mass of the satellite itself (\( m_s \)) to determine its orbital velocity. The mass of the satellite cancels out in the equations of motion since it does not affect the gravitational force acting on the satellite relative to the planet.

Therefore, the factor that is not needed when calculating the velocity of a satellite orbiting a planet is:

the mass of the satellite.

What was the initial cause of the Anglo-Powhatan Wars? (1 point) Responses John Rolfe's kidnapping of Pocahontas John Rolfe's kidnapping of Pocahontas Rebecca's marriage to John Rolfe Rebecca's marriage to John Rolfe English settlement on the Chesapeake English settlement on the Chesapeake theft of the Powhatan's resources by the English

The initial cause of the Anglo-Powhatan Wars was primarily rooted in tensions between the English settlers and the Powhatan Confederacy, which included conflicts over land, resources, and trade.

Among the options provided, the most accurate response would be:

theft of the Powhatan's resources by the English.

While events like John Rolfe's marriage to Pocahontas contributed to temporary peace and affected relations, the wars were fundamentally sparked by the encroachment of English settlers on Powhatan lands and their exploitation of resources. The English settlement in the Chesapeake region exacerbated these tensions, leading first to skirmishes and ultimately to open conflict.