This book struck me as struggling to be much more, or much less, as if somewhere there was a better book trying to get out. The first 100 or so pages (about 1/3 of the text) is (to me) a somewhat cursory history of rocket development from ancient China through Robert Goddard, Hermann Oberth, to Werner von Braun and the German V-2 work. I could have done without this, and you probably can, too, if you've read any prior histories of space exploration. This material felt like it was tacked as a publisher's suggestion to add pages.
The meat of Dickson's research, the year or so leading up to the USSR's Sputnik launch, with the eventual matching achievement by the US, is very well done, bringing out several important points. First, the frustration of the US team at not being allowed to send up a satellite earlier, which fed and was a result of inter-service rivalries. Second, the differences between the US and Soviet approaches (US - technologically more advanced, focused on multiple objectives with each launch; Soviet - based on somewhat simpler technology, generally one goal per launch; but see recent articles on Soviet rocket engines, which achieved much higher thrust for weight of fuel expended, and still do.) And finally, the apparent complacency of the Eisenhower administration at being "beaten" by the USSR, which actually pleased the president as it established the open skies policy meaning the US could overfly the Soviet Union at will for satellite reconnaissance, leading to the important CORONA series of launches spanning decades.