How Fast is an Asteroid Traveling Toward Earth?

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SUMMARY

The asteroid traveling toward Earth is calculated to have a speed of approximately 12.915 km/s when considering its average distance from the Sun of 1.9 AU. The formula used for this calculation is V² = G * M * ((2/r) - (1/a)), where G * M equals 1.327 * 10²⁰. The calculations reveal that at its furthest distance of 2.8 AU, the asteroid's speed can reach up to 36.162 km/s. The rocket, traveling at 12 km/s, must achieve a combined impact speed of at least 40 km/s for effective collision.

PREREQUISITES
  • Understanding of gravitational physics and orbital mechanics
  • Familiarity with the concept of astronomical units (AU)
  • Proficiency in using the formula V² = G * M * ((2/r) - (1/a)) for velocity calculations
  • Knowledge of SI units and unit conversion, particularly from kilometers to meters
NEXT STEPS
  • Research the implications of asteroid impact velocities on Earth and potential mitigation strategies
  • Study the effects of varying distances in elliptical orbits on celestial body speeds
  • Explore advanced gravitational physics concepts, including the three-body problem
  • Learn about the engineering challenges of intercepting fast-moving celestial objects
USEFUL FOR

Astrophysicists, aerospace engineers, students studying orbital mechanics, and anyone interested in planetary defense strategies against asteroid impacts.

fleetingmoment
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Homework Statement


A asteroid is hurtling towards Earth and humankind has decided to fire a nuclear warhead at it in order to avert disaster. In order be most effective the rocket carrying the warhead has to impact the asteroid at 40km/s. The rocket itself travels at 12km/s. What remains is to calculate the speed of the asteroid. During its elliptical orbit, the asteroid's greatest distance from the sun is 2.8 astronomical units (AU) and its smallest 1.00 AU. Its average distance from the sun is then 1.9 AU.

Homework Equations



1 x AU = 1.4960 * 1011m[/B]
The formula provided for calculating the speed of the asteroid is:
V2 = G * M * ((2/r) - (1/a))
where G *M = 1.327 * 1020 (gravitational constant times solar mass), r is the asteroid's distance from the sun (the book doesn't specify whether it is the greatest distance or the smallest) and a is its average distance from the sun.
2.8 * 1.4960 * 1011 = 4.1888 * 1011 = r
1.9 * 1.4960 * 1011 = 2.8424 * 1011 = a

The Attempt at a Solution


Plugging the relevant values into the equation thus:
V2 = 1.327 * 1020 * ((2/4.1888 * 1011) - (1/2.8424 * 1011))
gives 1.668 * 108
Taking the square root of both sides gives:
sqrt(V2) = sqrt(1.668 * 108) ⇔ V = 12915.1

Assuming my answer is correct, I've no idea what the given units are. Whether metres per second, or kilometres per hour, the value still seems incredibly high, given how fast asteroids actually travel. Have I gone wrong somewhere?
 
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Because G and M are SI units. All units must be SI units.
fleetingmoment said:
1 x AU = 1.4960 * 108km
km is not SI unit, you must change into meter
 
Hamal_Arietis said:
Because G and M are SI units. All units must be SI units.

km is not SI unit, you must change into meter
Thanks, Hamal_Arietis
The new value of 12915.1 seems a lot more realistic. Assuming it's also in metres per second. I'm going to divide by 1000 and conclude that the asteroid is traveling at 12.915 km/s.
 
If you find the asteroid velocity for both its nearest and furthest position from the Sun you will see that there is a range of velocities. I suppose you could find a particular distance where the rocket's speed and asteroid's speed combine to make the optimum collision speed.
 
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gneill said:
If you find the asteroid velocity for both its nearest and furthest position from the Sun you will see that there is a range of velocities. I suppose you could find a particular distance where the rocket's speed and asteroid's speed combine to make the optimum collision speed.
Thanks, gneill,
I plugged in the value for 1 AU and got 36161.5 or 36.162 km/s. I was obviously hasty in concluding that the human race was doomed, based on the value for the farthest distance - especially since the question should have read 'impact the asteroid at at least 40km/s': something I missed when translating the question from the language I'm studying in.