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

In summary, the question is asking for the minimum angle that the asteroid must be deflected in order to miss the Earth. Given the mass of the asteroid, its velocity, and the thrust generated by a rocket, the question is part of a larger problem that also asks for the time until impact and whether the Earth is saved. A similar question in the book had a different answer, which may indicate a calculation error.
  • #1
x252
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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.
 
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  • #2
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?
 
  • #3
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.
 

1. What is the angle between asteroids and Earth?

The angle between an asteroid and Earth varies depending on the specific location and orbit of the asteroid. Some asteroids may have a very small angle, while others may have a larger angle. It is also important to note that the angle between an asteroid and Earth is constantly changing as both objects move through space.

2. Can asteroids collide with Earth at a certain angle?

Yes, asteroids can collide with Earth at any angle. However, the likelihood of a collision depends on the size and trajectory of the asteroid, as well as the angle at which it enters Earth's atmosphere. Asteroids with a larger angle of approach are more likely to burn up in the atmosphere, while those with a smaller angle may have a higher chance of impact.

3. How does the angle between asteroids and Earth affect their impact?

The angle between an asteroid and Earth can greatly affect the impact of the asteroid. A direct hit at a 90 degree angle would result in maximum destruction, while a glancing blow at a smaller angle may cause less damage. The angle also affects the speed and trajectory of the asteroid, which can impact the size and location of the impact crater.

4. Can the angle between an asteroid and Earth be predicted?

Yes, scientists are able to predict the angle between an asteroid and Earth by studying their orbits and using mathematical calculations. However, the accuracy of these predictions may vary as the orbits of both objects can be influenced by other factors such as gravitational pull from other celestial bodies.

5. Is there a specific angle at which asteroids are more likely to hit Earth?

There is no specific angle at which asteroids are more likely to hit Earth. The likelihood of an asteroid impact depends on various factors such as the size, speed, and trajectory of the asteroid, as well as the angle at which it enters Earth's atmosphere. However, scientists are constantly monitoring and tracking potential asteroid threats in order to better understand and predict their paths and potential impacts with Earth.

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