You're tour director for a lunar trip, and want to award your passengers with certificates commemorating their crossing the point where the Moon's gravity becomes stronger than Earth's.How far from Earth should you award the certificate? Express your answer in meters.Express your answer as a fraction of the center-to-center Earth-Moon distance.
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To determine the point where the Moon's gravity becomes stronger than Earth's, we need to find the distance where the gravitational forces from the Moon and Earth are equal.
The force of gravity between two objects can be calculated using the equation:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravity between the two objects,
G is the gravitational constant (approximately 6.67430 × 10^-11 N*(m/kg)^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, we can assume that the mass of the passenger is negligible compared to the mass of the Earth and Moon, so we can leave it out of the equation.
Let's use M1 to represent the mass of the Moon, M2 for the mass of the Earth, and r to represent the distance where the gravitational forces are equal.
For the Moon's gravity:
F = G * (M1 * m2) / (r^2)
For the Earth's gravity:
F = G * (M2 * m2) / (r^2)
Since we want the point where the Moon's gravity becomes stronger, we can set these two equations equal to each other and solve for r:
G * (M1 * m2) / (r^2) = G * (M2 * m2) / (r^2)
Canceling out common terms:
M1 = M2
Therefore, the mass of the Earth and Moon must be equal for their gravitational forces to be equal.
Given this information, we can conclude that the point where the Moon's gravity becomes stronger than Earth's is at a distance equal to the center-to-center Earth-Moon distance.
Hence, the certificate should be awarded at a distance equal to the center-to-center Earth-Moon distance.
To determine the distance from Earth where the Moon’s gravity becomes stronger than Earth’s, we need to calculate the point where the gravitational forces are equal.
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
F = gravitational force
G = gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2)
m1 and m2 = masses of the two objects
r = distance between the centers of the two objects
In this case, the two objects are the Earth and the Moon. We can assume the mass of the Earth (m1) and the mass of the Moon (m2) as constant values.
Setting the gravitational forces of the Earth and the Moon equal to each other:
G * (m1 * m2) / r^2 (Earth) = G * (m1 * m2) / r^2 (Moon)
The masses of the Earth and Moon cancel out, giving us:
r^2 (Earth) = r^2 (Moon)
Taking the square root of both sides:
r (Earth) = r (Moon)
This means that the distance from the center of the Earth to the point where the Moon's gravity becomes stronger than Earth's is equal to the distance from the center of the Moon to the same point. Hence, the answer is the same fraction we use to express the center-to-center Earth-Moon distance.
The average center-to-center Earth-Moon distance is approximately 384,400 kilometers or 384,400,000 meters. Therefore, the distance from Earth where the Moon's gravity becomes stronger than Earth's is:
384,400,000 meters.