Spacecraft Deep Impact - Comet's mass

[treverd]

Homework Statement

On July 4, 2005, the NASA spacecraft Deep Impact fired a projectile onto the surface of Comet Tempel 1. This comet is about 9.0 {\rm km} across. Observations of surface debris released by the impact showed that dust with a speed as low as 1.0 {\rm m/s} was able to escape the comet.

1)Assuming a spherical shape, what is the mass of this comet? ( The escape speed for an object at the surface of Earth is 11.2 {\rm km/s} ).

2)How far from the comet's center will this debris be when it has lost 70 {\rm \%} of its initial kinetic energy at the surface?

I have absolutely no idea what to do. any help would be appreciated.

Homework Equations

I know that for part a it is V=sqrt((2GM)/r)

but then how I calculate the lost kinetic energy.

The Attempt at a Solution

 

[Hootenanny]

but then how I calculate the lost kinetic energy.

You know that it is 70% of it's intial value...

 

[Moetasim]

1) To calculate the mass of the comet, we can use the formula for escape velocity, which is given by V=sqrt((2GM)/r), where V is the escape velocity, G is the gravitational constant, M is the mass of the comet, and r is the radius of the comet (which is given as 4.5 km, since the comet is 9.0 km across and we are assuming a spherical shape). We can rearrange this equation to solve for M, which gives us M=(V^2*r)/2G. Plugging in the values, we get M=(11.2 km/s)^2*(4.5 km)/(2*6.67*10^-11 N*m^2/kg^2), which gives us a mass of approximately 6.06*10^13 kg for the comet.

2) To calculate the distance from the comet's center at which the debris will have lost 70% of its initial kinetic energy, we can use the formula for kinetic energy, which is given by KE=1/2*m*v^2, where KE is the kinetic energy, m is the mass of the debris, and v is its velocity. We can also use the conservation of energy principle, which states that the initial kinetic energy must equal the final kinetic energy plus the final potential energy. Since the debris is escaping the comet's gravitational field, its final potential energy will be zero. Therefore, we can set the initial kinetic energy equal to the final kinetic energy, and solve for the distance at which the debris will have lost 70% of its initial kinetic energy.

To do this, we can set up the following equation:

1/2*m*(1.0 m/s)^2 = 1/2*m*(v)^2

Where v is the velocity at the distance we are trying to find, and m is the mass of the debris (which we can assume to be the same as the mass of the comet, since the debris is very small compared to the comet). We can then solve for v, which gives us v=sqrt((1.0 m/s)^2*2), which is equal to approximately 1.41 m/s.

Now, we can plug this value for v into the equation for escape velocity, and solve for the distance (r) at which the debris will have lost 70% of its initial kinetic energy. This gives us r=(v^2*r