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Help needed interpreting the speed of an asteroid

  1. Oct 28, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

    1 x AU = 1.4960 * 1011m

    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
    3. 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?
     
    Last edited: Oct 28, 2016
  2. jcsd
  3. Oct 28, 2016 #2
    Because G and M are SI units. All units must be SI units.
    km is not SI unit, you must change into meter
     
  4. Oct 28, 2016 #3
    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 travelling at 12.915 km/s.
     
  5. Oct 28, 2016 #4

    gneill

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    Staff: Mentor

    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.
     
  6. Oct 28, 2016 #5
    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.
     
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