How did the Space Race grow out of the Cold War?

After the Second World War, two giants were left standing: the United States and the Soviet Union. They didn't trust each other one bit. But instead of fighting directly, they spent decades glaring across the globe, each trying to prove its way of life was best. People called this long stare-down the Cold War — cold because the two never quite came to blows.
A staring contest needs something to brag about. Both sides built bigger armies, bigger factories, bigger everything — anything to say, "Look how strong WE are." It was less a war and more a never-ending competition. And the best prizes were the ones the whole world could see.
