16 BEST AND MOTION 



with an acceleration, relative to the earth's surface, of 32.09 feet per second 

 per second. 



4. Supposing that the moon describes a circle of radius 240,000 miles round 

 the earth in 29 days, find its acceleration towards the earth. 



5. Assuming that the planets describe circles round the sun with different 

 periodic times, such that the squares of the periodic times are proportional to 

 the cubes of the radii of the circles, show that the accelerations of the planets 

 are inversely proportional to the squares of their distances from the sun. 



VECTORS 



13. We have found three kinds of quantities, motion, velocity, 

 and acceleration, all of which can be compounded according to 

 the parallelogram law. 



Quantities which can be compounded according to the parallelo- 

 gram law are called vectors. A vector must have magnitude and 

 direction, and hence must be capable of representation, on an 

 assigned scale, by a straight line. We have seen that motion, 

 velocity, and acceleration are all vectors. 



Composition and Resolution of Vectors in a Plane 



14. By definition of a vector, two vectors can be compounded 

 into one, by application of the parallelogram law. It also fol- 

 lows from the definition that any one vector may be regarded as 

 equivalent to two, these two being represented by the edges of a 

 parallelogram constructed so as to have the original vector repre- 

 sented by the diagonal ; or, as we shall say, 

 any vector can be resolved into two others. 



In particular, if we construct a rectangu- 

 lar parallelogram so as to have a line which 

 represents a vector R as its diagonal, we find 

 that the vector R can be resolved into two 

 vectors R cos e and R sin e, at right angles to one another, and in 



7T" 



directions such that R makes angles e, e with them. 



2i 



If we take two fixed rectangular axes Ox, Oy in a plane, we see 

 that any vector R can be resolved into two components R cos e, 



