There are 3 continuity errors when Billy takes a ride with Alan Shepard:
1. The plane on the tarmac has a different tail number than the plane in flight.
2. The plane on the tarmac has no name under the rear seat's canopy but the plane in flight does.
3. When Billy makes a left turn the exterior shot is flopped to make the plane's right turn appear to be a left turn (the tail number is mirrored).
The capsule with Mac in it shows a window, when in fact even Alan Sheppard didn't have a view port in his capsule. Gus Grissom was the first to have a view port in his Mercury flight.
The space flight is to be only 15 minutes, up and back down. The large map in the control room during the flight shows three orbits plotted.
When the boy is replacing the fuel line on the rocket he puts the damaged line back on.
Alpha is commanded to pick out a white triangle, but he actually picks up a white hexagon.
INCORRECTLY REGARDED AS GOOF: The T-38s shown in the film (NOT F-5s as previously incorrectly identified) did indeed enter service in the USAF as a trainer in March of 1961.
The saboteurs are parked outside Billy's house and follow him when takes off on his bike with a 1962 Chevy Impala. The scene is set in January, 1961.
One of the characters drives a Volkswagen, at least a 1958. VW came out
with square back window with the 58 model. This time period is the middle
of 1957.
The astronauts are shown flying F-5s and Alan Shepherd takes Willie for a ride in an F-5. The time period of this movie was 1960. The F-5 didn't enter service until 1962.
Although set in Cocoa Beach, Florida, Billy's jet ride with Alan Shepard lands at what is clearly Edwards Air Force base in California. There are no mountains in Florida, yet the air strip background clearly shows a Mojave mountain range.