12 The gravitational acceleration on the surface of a planet of radius \( R \) is \( g \). The gravitational acceleration at a height of \( 3 R \) above the surface of the same planet is: A \( \frac{\mathrm{g}}{16} \) B \( \frac{9}{9} \) C 16 g D 9 g

To find the gravitational acceleration at a height of \( 3R \) above the surface of a planet, we can use the formula for gravitational acceleration, which states that \( g' = \frac{g}{(1 + \frac{h}{R})^2} \), where \( h \) is the height above the surface. Here, \( h = 3R \), so: \[ g' = \frac{g}{(1 + \frac{3R}{R})^2} = \frac{g}{(1 + 3)^2} = \frac{g}{4^2} = \frac{g}{16} \] Therefore, the answer is **A \( \frac{g}{16} \)**. Now that we've resolved the question, let’s chat more about some cool facts and practical implications of gravitational acceleration! Did you know that gravitational acceleration varies not just with distance from a planet's center but also with the planet's mass and size? For example, while Earth has a gravitational acceleration of approximately 9.8 m/s², other celestial bodies like the Moon or Mars have much weaker gravitational pulls, which is why astronauts can jump so high on the Moon! In real-world applications, understanding gravitational acceleration is crucial for aerospace and engineering fields. Rocket launches take advantage of the varying gravity as they ascend through different layers of the atmosphere. Engineers must calculate the gravitational forces to ensure proper payload delivery to satellites and even other planets!