Washington — It was 1957, the year the Russians launched Sputnik I, the first space satellite, and the United States had been caught napping as Perry Como crooned the hit ''Dream Along With Me.''
Suddenly, the Space Race was on. America scrambled to catch up with the Russians, succeeding beyond its most moonstruck dreams. The past quarter century of patriotic space blitz has been captured in a new show here at the National Air and Space Museum: ''Twenty-five Years of Space Exploration.''
The new exhibit opens at a time when the nation is still fresh from celebrating its latest triumph, the landing of the space shuttle Columbia, complete with presidential welcome, on a sun-seared runway in the Mojave Desert.
The new show, which will run eight months, has landed in the museum's Milestones of Flight Gallery. Above it hovers the Wright Flyer, a frail, tan history-blazer that took off from Kitty Hawk, N.C., in 1903. Off to the left is the silvery gray Spirit of St. Louis in which ''Lucky Lindy'' Lindbergh made his historic transatlantic 1927 flight.
You hear the exhibit almost before you see it: strains of ''Jailhouse Rock'' roll by as Elvis Presley, dressed in convicts' stripes, wails his way through 1957. He is part of the first in a series of TV ''space capsules'' illustrating significant periods of the space generation. The time capsules are part of large visual displays, and come complete with countdowns till the programs begin.
All are witty, vivid looks at the particular time and culture in which achievements were made in space. For 1957-58, for instance, there are montages of backyard bomb shelters, Presley, ''droopy'' fashions, Eisenhower on the golf links, ''West Side Story,'' McDonald's hamburgers, test pilot John Glenn, gas at 23 cents a gallon, and a newspaper banner headline on the first abortive US attempt to send up a satellite: ''Oh, What a Flopnik.''
There's plenty on space, too, from footage of the historic Russian launching on Oct. 4, 1957 that shocked the US out of its scientific complacency, to Wernher von Braun, jaw clenched, promising we could launch a satellite in 90 days as the US did with the successful Explorer I on Jan. 31, 1958.
The show is designed to put American adventures in space into context with the times in which they happened, and to make the sequence clear, even to children. The museum is always aswarm with kids, like the blond nine-year-old who gazed at footage from the Vietnam era and said, ''I heard about that war. My father was in it.''
''Kids think we've always been in space, the way they think we've always had 'Big Macs,' '' says the man who put the exhibit together. He is Allan Needell, associate curator of the space science and exploration department at the museum. Mr. Needell says, ''I didn't want to trivialize the exhibit, didn't want to caricature American culture . . . but I included things like Elvis Presley and Big Macs because I didn't want to make it a space-only thing. There are book and record displays, too. . . . It's also about technological change and the way it affects our lives.''
So among the project displays for 1977-82 are John Cheever's collected short stories, Marilyn French's ''The Women's Room,'' Pachelbel's Canon in D Major (the theme of ''To Fly,'' the museum's film hit), and a pocket-size computer for Baryshnikov, ''Roots,'' Ayatollah Khomeini and the return of American hostages from Iran, as well as the Voyager spacecraft encounter with Jupiter and a look at the space shuttle.
The exhibit is broken down into five sections: The Beginnings (1957-58) on the early satellite period; Man in Space (1961-64), focusing on the Mercury flights of astronauts Alan Shepard and John Glenn; Lunar Conquest (1966-69), on the race to the moon; Space is Commonplace (1972-76), including Pioneer 10 headed for Jupiter as well as Skylab and the Apollo-Soyuz joint mission. The Today and Tomorrow unit (1977 to the present) focuses on Voyager and the space shuttle. Needell wrote the concept and scripts for the show, produced by Peter Vogt & Associates of Washington.
The exhibit is set in the middle of the museum, where a tiny slice of moon rock is embedded in a brass display case. The satiny gray triangle of rock, which draws a thick crowd of tourists, is a piece of basalt 4 million years old. ''I thought it was a piece of glass,'' a wide-eyed little girl said as she patted the lunar rock.
Also on display in the same area are the backup spacecraft for Explorer I, a model of the Sputnik I satellite, and the Apollo 11 command module. After visitors are through with the current exhibit, a 45-minute ''self-guided tour'' of the space collection is mapped out for them. The Smithsonian's two-story National Air and Space Museum, which Needell points out is the most popular museum in the world, has drawn 8 million visitors so far this year.