A 4.9 x 10^10 kg asteroid is heading directly toward the center of the earth at

Question

A 4.9 x 10^10 kg asteroid is heading directly toward the center of the earth at a steady 22 km/s . To save the planet, astronauts strap a giant rocket to the                 asteroid perpendicular to its direction of travel. The rocket generates 5.0 x 10^9 N of thrust. The rocket is fired when the asteroid is 4,2 x 10^6 km away from                 earth. You can ignore the rotational motion of the earth and asteroid around the sun.
                A
                If the mission fails, how many hours is it until the asteroid impacts the earth?
                Express your answer using two significant figures.
                B
                The radius of the earth is 6400 km . By what minimum angle must the asteroid be deflected to just miss the earth?
                Express your answer using two significant figures.
                                                                   

Explanation / Answer

The asteroid is traveling at 20 km/sec. To calculate how long it would take to reach the Earth if the mission fails, simply divide the distance it must travel by the speed (we're neglecting the radius of the Earth in this calculation because it is much smaller than the total distance):

t = D/v = (4.2*10^6 km)/(22 km/sec)
t = 1.9 * 10^5 sec = 52.7 hr

(b) Draw a picture of the situation. The center of the Earth will be at one corner of a right triangle, the "edge" of the earth will be at another, and the asteroid at third corner. The angle, theta, through which the asteroid must be deflected (neglecting the diameter of the asteroid itself) is given by:

tan(theta) = 6400 km/(4.2*10^6 km)
tan(theta) = 1.523*10^-3
theta = arctan(1.523*10^-3)
theta = 1.523*10^-3 radians (for small angles, tan(theta) ~= theta)

theta = 8.72 * 10^-2 degrees

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