Relationship of a planet's mass, size, and acceleration due to gravity

[yoosnb]

Homework Statement

Planet A and Planet B have the same mass, but planet A is twice larger than planet B. A ball dropped above the surface of planet A has an acceleration due to gravity of 10 m/s^2. which of the following is true if a ball is dropped 100 m from the surface of planet A.

Relevant Equations

A. If the ball is dropped 100 m from the surface of planet B, it will reach the ground at the same length of time it does at planet A.

B. at 1 s of its fall, the speed of the ball at planet A is less than the speed of the ball at planet B

C. the acceleration of the ball at planet A exceeds the acceleration of the ball at planet B.

D. the distance traveled by the ball at planet A is twice the distance traveled by the ball at planet B.

Choice D is obviously wrong therefore leaving us with choices A, B, and C. Can someone explain the relationship of the three variables stated above (mass, volume, and acceleration due to gravity)? Thank you.

 

[BvU]

Hello yoos, !

The whole idea of PF is that we help you with questions and hints, but you come up with the relevant equations and an attempt at solution yourself. So: what relationships have you learned so far between

• radius of a sphere and volume
• volume and mass
• mass, distance to center and gravitational acceleration

?

 

[yoosnb]

Hello,

Can the formula for Gravitational force between two objects be used in this scenario?

If yes, then I may have something in mind.

G_f = (G*m₁*m₂)/r²

Since mass is constant, we can disregard the numerator, leaving us with

G_f = 1/r²

Making the acceleration due to gravity inversely proportional to its radius or size. Therefore the answer is B.

Is this a valid solution/explanation for the problem?

 

[BvU]

Therefore the answer is B.

Don't see the logic ...

G_f = 1/r²

Never, never, never write something like that. It's dimensionally wrong and at some point it will bite you hard.Calculations/physics arguments in reasoning are possible for the cases (you sure there is only one correct answer ?)

e.g. A: B has a higher $g \Rightarrow$ A is wrong.

 

[Delta2]

I believe you got it right that B is the correct option, however your reasoning might not be entirely correct. It seems though that you got the core idea that is that the force varies with inverse square of the distance from the center of the planet.

To prove it correctly one way is to consider the ratio $\frac{F_A}{F_B}$ that is the ratio of the force the ball experiences at planet A (at the surface of the planet or at height 100m) ,$F_A$, to the force that experiences at planet B $F_B$ which ratio of forces is equal to the ratio of accelerations $\frac{g_A}{g_B}$ at planet A and planet B.