On the ball
To be alert; in command of one's senses.
Some authorities have suggested that 'on the ball' originated in the sporting arena, and alludes to runners being on the balls of their feet, eagerly ready to run a race. This has some similarities with being 'up to scratch', which derives from boxers or runners being ready at the starting line. It is a plausible derivation, but has nothing to recommend it beyond that.
A more commonly advocated location for the source of 'on the ball' is the Royal Observatory at Greenwich. This is where the oldest surviving and best known time-ball is sited. The Greenwich time-ball was installed in 1833 to signal the accurate time to passing ships. It was, and still is, raised just before 1pm each day and falls as 1pm strikes on the observatory's clock. Captains needed to have their ships' chronometers set accurately in order to navigate correctly, hence they needed to be 'on the ball'. It's a nice story and there are any number of tour guides around the observatory who are all too happy to repeat it. Unfortunately...
Need I go on? It isn't true.
The phrase 'on the ball' did actually originate in the sporting arena, but relates to the eyes rather than the feet. It is a contraction of the earlier expression 'keep your eye on the ball', which advice has been given to participants in virtually every known ball game. For the source, we need to look to early ball games. The phrase is recorded in early records of cricket, golf, croquet and baseball and many people regard baseball as the origin. Well, that appears to be almost true - the earliest citation that I can find in print comes from the English game of rounders. The English novelist William Kingston wrote 'books for boys', and in 1864 published Ernest Bracebridge, or, Schoolboy Days, which includes this scene:
Ellis seized the bat with a convulsive clutch... Remembering Ernest's advice, he kept his eye on the ball, and hit it so fairly that he sent it flying away to a considerable distance. "Capital!" cried Ernest. "Run! run! - two bases at least."
American readers will recognise the similarity of the rounders terminology with that of baseball. For those not familiar with rounders and/or baseball, suffice it to say that they are essentially the same game, but that it is easier to imagine Sylvester Stallone playing baseball. There's no consensus on this but there's a strong case to be made that baseball is in fact an English game, being merely a beefed-up variation of rounders.
In 1744, which is certainly before anyone is known to have played baseball in the USA, John Newbery, an English publisher and a man with a reasonable claim to be the originator of literature printed specifically for children, produced A Little Pretty Pocket-Book, intended for the Amusement of Little Master Tommy and Pretty Miss Polly. That title sounds entirely suitable as the source of the rules of the game of rounders, which is played nowadays by children. Nevertheless, the book includes a graphic labelled Base-Ball, which shows men playing the game and which is accompanied by a rhyme that pretty much sums up the basics of both rounders and baseball:
The ball, once stuck off,
Away flies the boy
To the next destin'd post,
And then home with joy.
Baseball may or may not have been the origin of 'keep your eye on the ball', but it did take over the use of the phrase. As well as as the batters 'keeping their eye on the ball', the pitchers were also said to 'put something on the ball', i.e. they imparted some spin or curve on it. This usage dates from the start of the 20th century, for example, this piece from The Indianapolis Star, April 1910:
Graham put something on the ball that fooled even Bowerman.
The figurative version of the phrase 'on the ball', i.e. with the meaning of being 'alert or apt' in a context where no actual ball is present, began later still. In 1989, W. C. Williams and J. Laughlin published Selected Letters, which contained an extract from a letter written Williams in 1939:
The novella by Quevedo... [is] right on the ball.
As to whether the phrase originated in the USA or the UK, on present evidence, I'd call it a 1-1 draw.