Finding Planetary Radius using density and escape velocity

  • #1
1. "Calculate the radius of a planet with mean density of 3.0x10^3 m2kg-3, from which a golf ball can be thrown to infinity as a velocity of 40 ms-1"


Homework Equations


I've been looking at the equation of:

Vesc=sqroot of 2*G*M/r and rearranging to r=2*G*M/Vesc. However, the trouble is, I'm struggling to get the mass from the density? It appears I don't have enough information, or I'm barking up the wrong tree so to speak?


The Attempt at a Solution



can anyone point me in the right direction please?
 

Answers and Replies

  • #2
Curious3141
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1. "Calculate the radius of a planet with mean density of 3.0x10^3 m2kg-3, from which a golf ball can be thrown to infinity as a velocity of 40 ms-1"


Homework Equations


I've been looking at the equation of:
fra
Vesc=sqroot of 2*G*M/r and rearranging to r=2*G*M/Vesc.


Recheck your algebra there. What happened to the square root?

However, the trouble is, I'm struggling to get the mass from the density? It appears I don't have enough information, or I'm barking up the wrong tree so to speak?


The Attempt at a Solution



can anyone point me in the right direction please?

Start with the conservation of energy statement ##\frac{1}{2}mv_e^2 = \frac{GMm}{r}## that leads to the equation you started with. ##m## is the mass of the golf ball (cancels out), while ##M## is the mass of the planet.

Now find an expression for ##M## in terms of the density and the radius. Assume the planet is a spherical ball of radius ##r##. What's the enclosed volume of a perfect sphere?
 
  • #3
Recheck your algebra there. What happened to the square root?



Start with the conservation of energy statement ##\frac{1}{2}mv_e^2 = \frac{GMm}{r}## that leads to the equation you started with. ##m## is the mass of the golf ball (cancels out), while ##M## is the mass of the planet.

Now find an expression for ##M## in terms of the density and the radius. Assume the planet is a spherical ball of radius ##r##. What's the enclosed volume of a perfect sphere?

The enclosed volume of a perfect sphere is is V=(3/4)∏*r3, and when rearranging the previous equation I get r=GM/0.5Ve^2 and M=G/0.5V^2*r?

I'm still struggling to see what I can do with this, as it appears that to find r I need M and to find M I need r? I'm really confused :-(
 
Last edited:
  • #4
Curious3141
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The enclosed volume of a perfect sphere is is V=(3/4)∏*r3, and when rearranging the previous equation I get r=GM/0.5Ve^2 and M=G/0.5V^2*r?

I'm still struggling to see what I can do with this, as it appears that to find r I need M and to find M I need r? I'm really confused :-(

You have ##V##. What's the relationship between mass, density and volume? Hence what is ##M## in terms of ##r##?

Replace ##M## with that expression. Rearrange to isolate ##r## on one side of the equation. That's just simple algebra. But be careful with it - you seem prone to making mistakes with this. The expressions you wrote are ambiguous (you should use LaTex formatting), but there seems to be mistake with the rearrangement here too.
 
  • #5
SteamKing
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The enclosed volume of a perfect sphere is is V=(3/4)∏*r3, ... I'm really confused :-(

Yes, you are confused.

Is your planet a flabby sphere? You should recheck your formula for the volume of a sphere. I also didn't understand the units of average density in the OP for the planet. The units of density are ML^-3.
 
  • #6
Yes, you are confused.

Is your planet a flabby sphere? You should recheck your formula for the volume of a sphere. I also didn't understand the units of average density in the OP for the planet. The units of density are ML^-3.

Hi, sorry about the typo, i should have wrote (4/3)∏r^3

In regards to units of density, again that was a typo and should read 3*10^3 kg M^-3, which is what has been provided in the problem set.
 
  • #7
Curious3141
Homework Helper
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Hi, sorry about the typo, i should have wrote (4/3)∏r^3

In regards to units of density, again that was a typo and should read 3*10^3 kg M^-3, which is what has been provided in the problem set.

OK, so what's ##M##, as I asked in my post? You may represent the density by ##\rho##.
 

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