HAT IS THE YEAR OF MANUFACTURE OF AN IVER JOHNSON 5 SHOT 38 CALIBER TOP BREAK HAND GUN WITH A ONE PIECE WOODEN HANDLE AND A SERIAL NUMBER OF U809?

You could contact the company here. They may have that information.
https://iverjohnsonarms.com/contact.html

The Iver Johnson Cadet was offered as (the basic gun) as the Model 55 Target, and the 55A then the 55S Cadet and the 55S-A Cadet. The "parent" is discribed as: IVER JOHNSON MODEL 55 TARGET,Revolver,double-action,22LR,8-shot,4 1/2",6" bbls and 10 3/4"(6") 9 1/4"(4 1/2") over all length weight 24 oz.(4 1/2") 26os(6"); large frame,pull pin cylinder release with flash control front rim; recessed cylinder chambers;fixed sights,oversized molded one piece Tenite plastic target type grips; Does not have the "hammer the Hammer'action,matte blue finish; After 1958 cylinder is fluted hard chrome lined barrel introduced in 1955; dropped from production in 1979..the .38 caliber offering is designated as the Model 55S-A which is the same as the above except that: it was offered in .22LR,.32 S&W and .38 S&W(this is NOT a .38S&W Special) and as an 8-shot in .22, as a 5-shot in both .32 and .38. it is 7 1/2" oal w/ loading gate on the Right side AFter 1974/1975 this model is called"I J Cadet it was introduced in 1961 and dropped in 1979 Values: Vgood=$110..Good=$85 and Fair=$75. This data is from the Gun Digest Book of Modern Gun Values and more detailed Iver Johnson histroy can be had from www.gunshowbooks.com..check out Bill Goforth's Iver Johnson Arms&Cycle Works Firearms 1871-1993..Bill also frequents these forums and he along with Jim Hauff will likely correct any misinformation I may heve relayed as well as correct any ommisions.

My best advice: take it to a gun shop. They know all about guns