Finding Planetary Radius using density and escape velocity

[elDuderino81]

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?

 

[Curious3141]

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?

 

[elDuderino81]

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 :-(

 

[Curious3141]

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.

 

[SteamKing]

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.

 

[elDuderino81]

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.

 

[Curious3141]

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 $\rho$.