How Fast is an Asteroid Traveling Toward Earth?

[fleetingmoment]

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?

 

[Hamal_Arietis]

Because G and M are SI units. All units must be SI units.

1 x AU = 1.4960 * 10⁸km

km is not SI unit, you must change into meter

 

[fleetingmoment]

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.

 

[gneill]

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.

 

[fleetingmoment]

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.