An astronaut exploring a distant solar system lands on an...

An astronaut exploring a distant solar system lands on an unnamed planet with a radius of $3860 \mathrm{~km}$. When the astronaut jumps upward with an initial speed of $3.10 \mathrm{~m} / \mathrm{s}$, she rises to a height of $0.580 \mathrm{~m}$. What is the mass of the planet?

Key Concepts

Acceleration due to Gravity

The acceleration due to gravity (g) is a measure of the gravitational force experienced at the surface of a celestial body. It governs how quickly an object speeds up as it falls and can be deduced from the behavior of a projectile, such as the height reached in a jump, showing the influence of the planet's gravitation.

Kinematic Equations for Vertical Motion

This concept involves using the equations of motion for objects under constant acceleration. In problems such as a jump, the maximum height achieved can be related to the initial velocity and the acceleration (due to gravity) using kinematic formulas, which allow one to solve for the gravitational acceleration.

Universal Law of Gravitation

This law states that every two masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. It is expressed mathematically as F = GMm/r², where G is the gravitational constant. This principle is fundamental in linking the gravitational acceleration at the surface to the mass and radius of the planet.

Mass Determination of a Planet

By combining the acceleration due to gravity with the universal law of gravitation, one can determine the mass of a planet. Specifically, the formula g = GM/r² can be rearranged to solve for the mass M, showing how observations of local gravitational effects (like a jump) can be used to infer the mass of a distant celestial body.

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