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 km across...

asked by @yinb1 • over 1 year ago • Physics • 5 pts

...Observations of surface debris released by the impact showed that dust with a speed as low as 1.0 m/s was able to escape the comet.

I'm not sure on how to solve part B! All I know is that because 70% of its intial kinetic energy is lost, there is 0.3PE

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1 answer

At the surface of the comet, the debris has a potential energy due to gravity and a kinetic energy due to traveling at the escape velocity:

PEi + KEi = -GMm/R + (1/2)mv_esc^2

where I have chose M as the mass of the comet and m as the mass of the debris. At some distance d away from the center of the comet, the debris is going to have 30% of its initial kinetic energy (it lost 70%, so it has 30% left), and it will have a potential energy due to gravity:

PEf + KEf = -GMm/d + 0.3*(1/2)mv_esc^2

Note that the final distance is d, whereas the initial distance was R, whatever the radius of the comet is, and our final kinetic energy is just 0.3 times the initial kinetic energy (or 30% of the initial). Setting the initial energy equal to the final energy,

PEi + KEi = PEf + KEf

--> -GMm/R + (1/2)mvesc^2 = -GMm/d + 0.3*(1/2)mvesc^2

First of all, you can see that we can cancel m from all the terms (which is good because we don't know the mass of the debris),

--> -GM/R + (1/2)vesc^2 = -GM/d + 0.3*(1/2)vesc^2

Now, we can start plugging things in. First of all, the kinetic energy (divided by the mass m) at the escape velocity is

(1/2)v_esc^2 = (1/2)*(1.0)^2 = 0.5 J/kg <--- the units are J/kg b/c we divided by m

So, 30% of this value is

0.3(1/2)v_esc^2 = (0.3)(0.5) = 0.15 J/kg

Now, the potential energy initial (divided by the mass m) is (noting that the mass of the comet, given in part a, is 3.4x10^13 kg, and the radius is 3.5 km):

-GM/R = -(6.67x10^-11)*(3.4x10^13) / (3.5 x 10^3) = -(2267.8) / (3.5 x 10^3)

Note that GM = 2267.8, because we're going to need that as well,

-GM/R = - 0.648 J/kg <-- once again, it's J/kg because it's energy per mass

So, going back to our energy conservation equation,

-GM/R + (1/2)vesc^2 = -GM/d + 0.3*(1/2)vesc^2

--> -0.648 + 0.5 = -(2267.8) / d + 0.15

--> (2267.8) / d = 0.15 - 0.5 + 0.648 = 0.298

--> d = 2267.8 / 0.298 = 7,610 m

So, the distance between the center of the comet and the debris is 7.61 km.

answered by @doug • over 1 year ago