Force question involving an asteroid and spacecraft

[rkjul]

Homework Statement

This is a 5-part problem.

In June 1997 the NEAR spacecraft ("Near Earth Asteroid Rendezvous"; see http://near.jhuapl.edu/), on its way to photograph the asteroid Eros, passed within D=1200 km of asteroid Mathilde at a speed of 10 km/s relative to the asteroid (Figure 2.34). From photos transmitted by the 805 kg spacecraft , Mathilde's size was known to be about 70 km by 50 km by 50 km. It is presumably made of rock. Rocks on Earth have a density of about 3000 kg/m3 (3 grams/cm3).

(a) Calculate the mass of the asteroid, using the simplistic assumption that it has a rectangular cross-section.

(b) Calculate the magnitude of the force of gravity acting on the spacecraft due to the asteroid, when they are the distance D=1200000 m apart.

(c) For comparison, calculate the magnitude of the force of gravity acting on the spacecraft when it was on the surface of the Earth.

(d) Estimate the change in the spacecraft 's momentum due to its interaction with the asteroid, using the following method: Instead of the actual force of gravity which acts at all distances through the 1/r^2 force law, replace it with a simple impulse from a constant force acting over a fixed time, and which is zero for all earlier and later times. Assume there is no force on the spacecraft until it is very close. Then assume that the force is equal to the maximum value you calculated in part (b) above, and that it lasts for the time required to travel a distance D=1200000 m. You can safely assume that there is no significant deviation of the spacecraft from its original path with its original velocity during this time.

(e) Using your result from part (d), make a rough estimate of how far off course the spacecraft would be after 4 days.

Homework Equations

density=mass/volume
Fgrav= -G*[(m1m2)/(r)^2]
f=mg
deltap=Fnet*(deltat)
deltaposition=velocityavg*(deltat)

The Attempt at a Solution

Alright, I got the first 3 parts fairly easily by plugging and chugging into the first 3 equations. It took me a bit longer to figure out part d but I used 1200000m for change of position and set that equal to 10000m/s * deltat like this and solved for delta t in seconds:

1200000=10000*(deltat)

I came up with 120 seconds which I know is the correct answer and then I used that time in the equation for change of momentum and multiplied by the force I obtained in part b, 1.97e-2 N.

deltap=(1.97x10^-2)*120

I got 2.364 kg*m/s.

Part e is where I'm stumped. I'm not sure how to use the CHANGE in momentum to determine where the spacecraft will be as I don't know the final momentum of the spacecraft . Do I need to draw a right triangle for this part of the problem and figure out the answer that way?

 

[learningphysics]

Not sure... but I think maybe, assume the change in momentum is perpendicular to the original velocity. So this new velocity component (which you can get from the change in momentum you calculated) * time... equals distance off course?

 

[rkjul]

I got it by using deltap=mass*deltav and solving for deltav, then multiplying by deltat (4 days and doing appropriate coversion). I had been working on this assignment for a long time. It's funny how you can sometimes get a brain cramp over fairly simple things.