Two satellites are orbiting the Earth at heights h1 and h2. If h2 = 2h1, what is

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Question: Two satellites are orbiting the Earth at heights h1 and h2. If h2 = 2h1, what is the ratio of their orbital speeds?

Options:

Correct Answer: 1:√2

Solution:

v = √(GM/(R+h)), for h2 = 2h1, v2 = √(GM/(R+2h1)), thus v1/v2 = √(R+2h1)/(R+h1) = 1:√2.

Two satellites are orbiting the Earth at heights h1 and h2. If h2 = 2h1, what is the ratio of their orbital speeds?

Two satellites are orbiting the Earth at heights h1 and h2. If h2 = 2h1, what is the ratio of their orbital speeds?

Step 1: Understand that two satellites are orbiting the Earth at different heights, h1 and h2.
Step 2: Note that h2 is twice h1, so we can write h2 = 2h1.
Step 3: Recall the formula for orbital speed: v = √(GM/(R+h)), where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.
Step 4: For the first satellite at height h1, the orbital speed v1 is given by v1 = √(GM/(R+h1)).
Step 5: For the second satellite at height h2 (which is 2h1), the orbital speed v2 is given by v2 = √(GM/(R+2h1)).
Step 6: To find the ratio of their speeds v1/v2, we set up the equation: v1/v2 = √(GM/(R+h1)) / √(GM/(R+2h1)).
Step 7: Simplifying the ratio gives us v1/v2 = √((R+2h1)/(R+h1)).
Step 8: Now, we need to evaluate this ratio. Since h2 = 2h1, we can substitute and simplify further.
Step 9: The final ratio of their speeds is v1/v2 = 1:√2.

Orbital Mechanics – Understanding the relationship between the height of an orbit and the orbital speed of a satellite.
Gravitational Force – Applying the formula for gravitational force and its effect on orbital speed.
Ratios and Proportions – Calculating and simplifying ratios of speeds based on different heights.