How old were you at the time?
Maybe five?
That’s pretty young. Your memory of what you saw may be correct, but the little detail of how the ship was moving and rocking may not be right. Or you had a better toy than we found so far. Either way, I’m jealous that you had that thing as a kid.
I had a very similar board game in the late '80s. It was a huge board and each side had a submarine that shot little red discs at three ships with tiny elastics which would “explode” when the discs triggered them.
[Googles game] Ah, I found it. It was called Torpedo Run! Not a lot of strategy involved, but it was fun to shoot and blow up the ships.
I loved the hell out of this game back in the day.
Were those Crash Derby or something similar? You pulled the plastic cord, which spun a gear inside and off it went. You crashed them into other crash derby cars for increased destrcto-fun.
Keystone Exploding Ship w/ Atomic Submarine! This is the name of the toy that you are inquiring about! I had that same toy also! exploding battleship destroyed by sub with projectile!
Was the carpet burnt orange or avocado? 
Not motorized. Sorry.
I found this image, which looks older than what I had. Mine was grey plastic, and the target panel was rectangular. Both the ship and the sub looked more realistic, too.
I think it was ‘cream’.
Maybe it was when you were done with it.
I think I found it:
http://www.vectis.co.uk/Page/ViewLot.aspx?LotId=514995&Section=7051
Rectangular target, creme color and it pitched and rolled.
That’s it!
Thanks!
(NB: It was the carpet that I think was cream-coloured. )
Hah! If I had not made that mistake I would had searched and wasted a lot of time looking for other choices. As it turned out I looked for that kind of creme-gray color that I also remembered seeing in military toys from the past (I used to have a plastic aircraft carrier of that color).