Describe the Arms Race and the Space Race between the U. S. and the U. S. S. R.

I presume you mean the Nuclear Arms Race, so my answer is based off that:

The nuclear arms race was a arms race competition for supremacy in nuclear warfare between the United States, the Soviet Union, and their respective allies during the Cold War. During this period, in addition to the American and Soviet nuclear stockpiles, other countries developed nuclear weapons, though none engaged in warhead production on nearly the same scale as the two superpowers.

The Space Race:

The Space Race refers to the 20th-century competition between the Soviet Union (USSR) and the United States (US), for dominance in spaceflight capability. It had its origins in the missile-based nuclear arms race between the two nations that occurred following World War II, aided by captured German missile technology and personnel from the Aggregat program. The technological superiority required for such dominance was seen as necessary for national security, and symbolic of ideological superiority. The Space Race spawned pioneering efforts to launch artificial satellites, uncrewed space probes of the Moon, Venus, and Mars, and human spaceflight in low Earth orbit and to the Moon.