Transmission in a 1962 Ford Falcon

Without knowing the engine speed/gearing I could not say. The engine should not run over 5,000 rpm. Perhaps your car is stuck in first gear? Does it shift up at some point?

Yes it has first year and drive when you put it in drive it does shift once but when I get up around 50 55 it sounds like the motor is working real hard

Install a tachometer and see what the engine is actually turning. I would not want to run higher than 3 grand when cruising down the road. You could probably go as high as 5 grand when passing. Which engine do you have?

I had the same transmission in my falcon, I tended to drive it around 40-45, but I would stick to 40. Personally, if you want to keep the trans, I would change out the rear gearing, because it screamed the same way you described yours...

i have a 62 falcon with the 2 speed automatic and it sounds like it is running to fast after the gear change as well. Why would they make and sell the car if the transmission wasnt sufficient? i remember seeing one pass me on the highway going about 70. I drive mine now at about 50 when on the faster roads. Readying this exchange makes me afraid to go that fast now.