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Solution to Exercise 9

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motion: TOC for Knowledge Concepts, Exercises, and Solutions

In one hour, the asteroid moves 1.5 degrees.  Thus the number of radians it moves 
is 1.5 deg / 57.3 deg  = 0.026 radians.  Remembering that the angle of something 
in radians is equal to the ratio of its diameter to its distance, we know that 

     distance travelled in one hour            X
     ------------------------------  =  ----------------  = 0.026 
        distance to the asteroid          4,500,000 km

Thus a distance X = 117,000 km has been travelled by the asteroid in one hour.

Another way to solve this problem using trigonometry is to realize that 1.5 
degrees is a very small angle; thus, you can approximate the ratio of transverse 
length to distance as the tangent of a right triangle with angle 1.5 degrees.  In 
this case, 

    tan(1.5 degrees) = 0.026 = X / 4,500,000 km  and X = 117,000 km.

Thus its transverse (i.e. horizontal) speed, with respect to our viewpoint on 
Earth, is  117,000 km / 3600 s  = 32.5 km/s (your precision may vary).