How high does a rocket have to go above Earth's surface until its weight is one fourth what it would be on Earth?
To determine the height at which a rocket's weight is one fourth of what it would be on Earth, we can use the concept of gravitational force.
1. The weight of an object is the force of gravity acting on it. On Earth, the weight of an object can be calculated using the formula:
Weight = mass × acceleration due to gravity
2. The acceleration due to gravity on Earth is approximately 9.8 m/s².
3. Let's represent the weight on Earth as W₁ and the weight at the desired height as W₂. Given that W₂ is one fourth of W₁, we can write:
W₂ = 1/4 × W₁
4. Since weight is directly proportional to the mass of the object, we can say:
W₂ = 1/4 × W₁ = 1/4 × (mass × g)
5. Cancelling out the weight terms, we have:
1/4 × (mass × g) = mass × (g/R₁²) × R₂²
where R₁ is the radius of Earth, R₂ is the height above the Earth's surface.
6. Rearranging the formula, we get:
R₂² = R₁² × 1/4
7. Taking the square root of both sides, we have:
R₂ = R₁ × 1/2
Therefore, the height at which a rocket's weight would be one fourth of what it would be on Earth is half of Earth's radius. The average radius of Earth is approximately 6,371 kilometers, so the rocket would need to reach an altitude of 3,185.5 kilometers above the Earth's surface.
To find out at what altitude a rocket's weight is one-fourth of what it would be on Earth, you can use the concept of the inverse square law of gravity. According to this law, the weight of an object decreases inversely proportional to the square of the distance from the center of the Earth.
Here's how you can calculate it step by step:
1. Determine the weight on Earth: Let's assume the weight of the rocket on Earth is W.
2. Calculate the factor by which the weight decreases: Since the weight is one-fourth, the weight at the higher altitude will be W/4.
3. Set up the equation: Use the inverse square law of gravity equation: W/4 = W/(distance^2).
4. Solve for the distance: Rearrange the equation to solve for distance: distance = sqrt(4).
5. Calculate the distance: Calculate the square root of 4 to find the distance factor. The square root of 4 is 2.
6. Multiply by the radius of the Earth: The distance obtained represents the ratio of the higher altitude to the radius of the Earth. Multiply this ratio by the radius of the Earth to get the final distance.
The radius of the Earth is approximately 6,371 kilometers (or 3,959 miles).
Distance = ratio x radius of the Earth
Distance = 2 x 6,371 kilometers
Distance ≈ 12,742 kilometers (or 7,918 miles)
Therefore, a rocket would need to go approximately 12,742 kilometers (or 7,918 miles) above the Earth's surface for its weight to be one-fourth of what it would be on Earth.
sqrt (1/4)=1/2
so wouldn't it be one radius of earth above earth (doubling distance from center of Earth).