# Force question involving an asteroid and spacecraft

• rkjul
In summary, the spacecraft passed within 1200 km of asteroid Mathilde and was able to calculate its mass and gravity based on the data transmitted. The magnitude of the force of gravity acting on the spacecraft was calculated to be 1.97e-2 N. If the spacecraft were on the surface of Earth, its momentum would be decreased by this force. The change in momentum would cause the spacecraft to be off course by about 120 seconds after passing by the asteroid.
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?

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?

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.

## 1. What is the force exerted by an asteroid on a spacecraft?

The force exerted by an asteroid on a spacecraft is dependent on the mass and velocity of the asteroid, as well as the distance between the asteroid and the spacecraft. The force can be calculated using Newton's law of gravitation.

## 2. How does the force of an asteroid affect the trajectory of a spacecraft?

The force of an asteroid can cause a change in the velocity and direction of a spacecraft, altering its trajectory. This is known as the gravitational pull of the asteroid and can be used to slingshot a spacecraft towards its destination.

## 3. Can the force of an asteroid damage a spacecraft?

Yes, if the asteroid is large enough and the spacecraft is not equipped to withstand the impact, the force of the asteroid can cause significant damage. However, most spacecraft are designed to withstand impacts from small debris and asteroids.

## 4. How does the force of an asteroid on a spacecraft differ from the force of gravity on Earth?

The force of an asteroid on a spacecraft is similar to the force of gravity on Earth in that it is an attractive force between two objects. However, the force of gravity on Earth is a constant downward force, while the force of an asteroid can vary depending on the distance and mass of the objects involved.

## 5. Can the force of an asteroid be used to propel a spacecraft?

Yes, the force of an asteroid can be used to propel a spacecraft through the technique of gravitational slingshotting. This involves using the gravitational pull of an object, such as an asteroid, to change the velocity and direction of the spacecraft, allowing it to travel further and faster.

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