Question

1. scientists are trying to decide if earth or mars has a larger gravity well. if earth has a mass of 610^24 kg and mars has a mass of 6.3910^23 kg, which has a larger gravity well? explain your answer.

Explanation:

Step1: Compare the masses

We know that Earth has a mass of \(6\times10^{24}\text{ kg}\) and Mars has a mass of \(6.39\times 10^{23}\text{ kg}\).
We can rewrite \(6\times10^{24}\) as \(60\times10^{23}\).

Step2: Analyze the relationship between mass and gravity

According to the law of universal gravitation \(F = G\frac{Mm}{r^{2}}\) (where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(M\) and \(m\) are the masses of two objects, and \(r\) is the distance between their centers of mass). For objects on the surface of a planet (assuming \(r\) is related to the planet's radius and for a given planet - object system, \(G\), \(m\) (mass of the object on the planet) and \(r\) (planet's radius) are constant), the gravitational force (and thus the "gravity" experienced by an object on the planet's surface) is proportional to the planet's mass \(M\).
Since \(60\times 10^{23}>6.39\times 10^{23}\) (because \(60 > 6.39\) when the exponents of \(10\) are the same)

Answer:

Earth has a larger gravity. Because, when comparing the masses \(M_{Earth}=6\times 10^{24}\text{ kg}=60\times 10^{23}\text{ kg}\) and \(M_{Mars} = 6.39\times 10^{23}\text{ kg}\), and since gravity (for a given object on the planet's surface, with other factors like the gravitational constant and the ratio of the object's height to the planet's radius being relatively constant) is proportional to the planet's mass, a larger - mass planet (Earth in this case) has a larger gravity.