Help calculating mass and escape velocity Options

[Monkey]

If you have a kuiper belt object with say a diameter of 48km and assume a density of 1.5 gm/cm^3, how do you find the mass and escape velocity?

Sorry if this sounds elementary, I am just trying to learn. I think that the surface area would be 7,238km^2 (4pi(r^2)) and that the volume would be 57,905km^3 ((4/3)*pi*r^3). I seem to start getting into trouble solving for the mass (given that density = mass/volume.) Would the mass only be 1.5 * 57,905,000m^3 = 86,857,500m^3 = 86,857.5 kg? What would the escape velocity be?

[Greg Hullender]

I guess we're assuming this KBO is a sphere. :-)

The surface and volume calculations are correct; you get into trouble trying to use your density (which is g/cm^3) with your volume (which is km^3). Also, although there are 1,000 m in a km, there are 1,000,000 m^2 in a km^2 and 1,000,000,000 m^3 in a km^3. Since your density is in cm^3, you'll need to go from km^3 to cm^3 before you can multiply it by the density, and your result will be grams.

Escape velocity is sqrt(2MG/r) where M is the mass of the KBO, G is the gravitational constant, and r is the radius. You cannot derive this formula without Calculus, but if you do know Calculus, it's not hard. Again, you'll need to get the gravitational constant into the same sort of units as you're using for M and r.

Hope that helps.

So is this for a class, or just your own curiosity? :-)

--Greg

[Monkey]

I guess we're assuming this KBO is a sphere. :-)

The surface and volume calculations are correct; you get into trouble trying to use your density (which is g/cm^3) with your volume (which is km^3). Also, although there are 1,000 m in a km, there are 1,000,000 m^2 in a km^2 and 1,000,000,000 m^3 in a km^3. Since your density is in cm^3, you'll need to go from km^3 to cm^3 before you can multiply it by the density, and your result will be grams.

Escape velocity is sqrt(2MG/r) where M is the mass of the KBO, G is the gravitational constant, and r is the radius. You cannot derive this formula without Calculus, but if you do know Calculus, it's not hard. Again, you'll need to get the gravitational constant into the same sort of units as you're using for M and r.

Hope that helps.

So is this for a class, or just your own curiosity? :-)

--Greg

Hi Greg;

I am a freshman in high school, so no calculus yet. So if I am doing the mass calculation correctly:

Given a basically spherical KBO that is 48km in diameter with an assumed density of 1.5 gm/cm^3 and thus volume = 57,905km^3; Mass = 1.5 * 57,905km^3 = 86,857.5km^3. (km^3 conversion: 1000^3 = 10^9m; Further conversion to cm = another 10^2) so the mass = 8.68575 x 10^15kg? (9+2+4) or am I still off by a factor of 10-1000?

Help!