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Orbit Altitude: Astronauts Above Planet's Surface
Chapter 13, Problem 42P(choose chapter or problem)
A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet’s surface. How far above the surface are they orbiting? Your answer will be a multiple of R.
Questions & Answers
QUESTION:
A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet’s surface. How far above the surface are they orbiting? Your answer will be a multiple of R.
ANSWER:
Step 1 of 2
The force of gravity acting on an object is given by:
\[ F = G \cdot \frac{M \cdot m}{r^2} \]
where:
\begin{align*} G & \text{ is the gravitational constant,} \\ M & \text{ is the mass of the planet,} \\ m & \text{ is the mass of the object (in this case, the starship), and} \\ r & \text{ is the distance from the center of the planet to the object.} \end{align*}
The gravitational acceleration \( g \) at the surface of the planet is: \[ g = G \cdot \frac{M}{R^2} \] And the gravitational acceleration at the starship's altitude \( a \) is:
\[ a = G \cdot \frac{M}{(R + h)^2} \] We're given that: \[ a = \frac{g}{2} \]
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Orbit Altitude: Astronauts Above Planet's Surface
Want To Learn More? To watch the entire video and ALL of the videos in the series:
Embark on a celestial journey! In this video, we calculate the altitude above a distant planet's surface where a starship orbits. Discover how astronauts experience reduced free-fall acceleration and the fascinating dynamics of space travel.