What is the minimum angle required to deflect an asteroid heading towards Earth?

[x252]

Homework Statement

A 4.50E10 kg asteroid is heading directly toward the center of the Earth at a steady 16.0 km/s. To save the planet, astronauts strap a giant rocket to the asteroid perpendicular to its direction of travel. The rocket generates 5.0E9 N of thrust. The rocket is fired when the asteroid is 5.70E6 km away from earth. You can ignore the rotational motion of the Earth and asteroid around the sun.

The radius of the Earth is 6400 {\rm km}. By what minimum angle must the asteroid be deflected to just miss the earth?

Homework Equations

Trigonometric functions (more specifically, tan(x))

The Attempt at a Solution

Made a picture (pictorial representation?) of a right triangle, the adjacent angle being 5.7E6km, and the opposite being 6400km (the radius of the earth). Did tan(6400/5.7E6) and tan(128000/5.7E6), neither of these produced the correct answer. I'm guessing the problem would be easier if we were given the radius of the asteroid, which we aren't.

 

[tony873004]

You shouldn't need the radius of the asteroid, as it's insignificant compared to the radius of Earth. If they gave it to you, you'd simply add it to Earth's radius. But since Earth's radius is expressed to only 2 significant figures, it wouldn't make a difference.

tan(6400/5.7E6) should have worked. So should sin(6400/5.7e6) as the opposite and hypoteneuse are virtually identical. Is your calculator in Degree mode or Radian mode? What is the answer you're expecting to get? Why did they give you all that other info (mass of asteroid, velocity, and thrust)? Is this one part of a longer question?

 

[x252]

Hello tony!

Thank you for the quick reply.
My calculator is in degree mode.

That question is indeed part of a larger one (Preceded by "If the mission fails, how many hours is it until the asteroid impacts the earth?" with the answer 99.0 hours, and followed by the question "The rocket fires at full thrust for 202 s before running out of fuel. Is the Earth saved?", which I'm assuming why they gave the rest of the information.

A similar question in the book (our online homework is only slightly changed from the book) had an answer of far closer to 1degree.