Bdtyk

Though there are different arguments regarding the ball in the phrase 'on the ball'. Dictionary.com has the following definition which points towards the ball of any ball games and seems quite sensible:

get on the ball

verb phrase

To pay closer attention to doing something right; improve one's
performance •Often an exasperated command

[1940s+; fr keep your eye on the ball, fr baseball or other ball
sports]

---

The Dictionary of American Slang, Fourth Edition by Barbara Ann Kipfer, PhD. and Robert L. Chapman, Ph.D. Copyright (C) 2007 by HarperCollins Publishers.

The Phrase Finder states the following:

• 
  

  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.

  
  

• 
  

  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'.

  
  

• 
  

  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.

  
  

The ball in the earliest uses of the phrase "be on the ball" seems to have been literally a ball of one of the various sorts used in different sports; but over time it became figurative for "the task at hand" or "the thing of importance."

Paul Dickson, The Dickson Baseball Dictionary (1989) suggests that the broader idiom on the ball originated with baseball:

on the ball adj. Describing a pitcher who is working well. A good pitcher with the ability to deceive batters is said to have a lot on the ball. "Confidence is great stuff but the pitcher must put something else on the ball." (San Francisco Call, October 16, 1913; PT)

Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) agrees with Dickson about the origin of the term:

on the ball, be Also, have a lot or something on the ball. Be especially capable or efficient, as in These programmers really have a lot on the ball. This term originated in baseball, where it was used for throwing a pitch with exceptional speed, spin, or some other deceptive motion {Slang; early 1900s}

Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins (1997), however, offers a somewhat different analysis:

on the ball. Depending on how it is used, the expression has two different origins. To be on the ball, "to be alert, knowledgeable, on top of things," probably refers to close and clever following of the ball by players in British soccer or American basketball. The phrase may have arisen independently in each sport, or it may have originated in the 1940s with the "bop and cool" jazz musicians and fans as the American Dictionary of Slang suggests. There is no hard evidence for any theory, but the sports analogy seems more logical. To have something on the ball, "to be talented or effective in some way," is surely of American origin, a baseball term referring to the various"stuff"—curves, spin, etc.—a good pitcher can put on the ball to frustrate a batter.

I ran an Elephind search for "to be on the ball" and found this instance from "Berserkers v. Wanderers," in the [Rockhampton, Queensland] Morning Bulletin (July 21, 1891):

"Doctor" was one of the best of tho backs, always coming to the rescue with a capital run and kick. R. Thomas and Bree also played particularly well. Among the forwards J. Thomas, T. M'William, W. Schmidt, and H. B. Gavin worked with a will, and always seemed to be on the ball. For the losers H. Carpendale played in grand form, kicking, running, and tackling splendidly. On two or three occasions he was within an ace of scoring, but was forced into touch in the nick of time.

A similar instance appears in "Sporting," in the [Broken Hill New South Wales] Barrier Miner (May 15, 1893):

That Mr. A. Trenberth is a success as central umpire there can be no doubt. His speed enables him to be on the ball throughout the match, he is using his influence to discourage wing play, and his decisions seem very just. It must have been in a moment of indecision that in the second quarter of Saturday's match he acted in a manner which mystified onlookers and players alike.

This last instance is significant because, whereas one might describe a football or basketball players as being "on the ball" because of being in direct continuous or repeated contact with the ball, a referee never intentionally comes in contact with the ball during play. So in this case "on the ball" wins out over the more descriptively accurate "near the ball" and may represent an early stage of idiomatic usage of "on the ball."
Australians, at any rate, seem to have been using the phrase in Hendrickson's "non-baseball" sense long before the 1940s.

RAILROAD WATCHES at the end of the 19th Century and well into the first half of the 20th Century were depended on to run trains on time. In the late 1890s there was a very bad train wreck which took place because a railroad conductor's pocket watch was off-time by more than eight minutes; based on what he BELIEVED was the correct time, the conductor gave the go-ahead to his train's locomotive engineer to proceed along a stretch of single track, relying on the schedule that showed that no trains would be traveling in the opposite direction along that same stretch of single track at that time. As a result of this conductor's inaccurate watch, his train collided head-on with another train traveling in the opposite direction, resulting in numerous deaths and injuries.

After this fatal wreck, railroad timekeeping, dependent on the precision, accuracy and reliability of railroad pocket watches, had to be raised to a much higher standard, wherein it became mandatory that ALL railroad watches in the United States had to be regularly inspected for accuracy and reliability, and removed from service for repair when found to be faulty.

The person put in charge of establishing and enforcing these new rules and setting the new higher minimum standards a railroad pocket watch had to meet in order to qualify as such, was a man by the name of Webb C. Ball, of Cleveland, Ohio, who was at the time the general time inspector for over 125,000 miles of railroad in the U.S.A., Canada and Mexico.

The American watch manufacturers Elgin, Hamilton, Waltham and others were therefore mandated to raise their standards for accurate time-keeping to meet Ball's new requirements. Some railroad watches were manufactured and sold under Webb C. Ball's own name, and Ball's 23-jewel railroad pocket watches have engraved on their dial faces, "BALL - OFFICIAL STANDARD - CLEVELAND." Thus it came to be that the phrase, "to be on the BALL," in railroaders' lexicon, meant that "YOUR TRAIN IS RUNNING ON TIME."

This story of railroad time-keeping adds yet another curious dimension to the many equally relevant sports- and maritime-related concepts of what it means "to be on the ball."

As a shortening of "keep your eye on the ball", baseball and golf come to mind. The meaning implies focus and alertness. There is also the phrase "behind the 8-ball", from billiards, used metaphorically to describe a sort of final moment, the last shot that precludes victory.

It's the ball of your foot:

on which the weight of the body rests when the heel is raised.

giving you balance and poise.

Evidence: poise (from Online Etymology Dictionary)

The sense of "steadiness, composure" first recorded 1640s, from notion of being equally weighted on either side (1550s). Meaning "balance" is from 1711; meaning "way in which the body is carried" is from 1770.

I think of being 'on the ball' more like balancing oneself on an actual ball, like an elephant in the circus, or an idiot at the gym. To be on the ball would mean having the balance or foresight to identify and fulfill needs before they are needed.

If a nurse hands a doctor a scalpel before he has time to ask for it, the nurse can be said to be 'on the ball.'

If a camper pulls out a bag of marshmellows that somebody else was supposed to bring, they are 'on the ball'

Balancing on a ball is similar, in that one has to have the awareness and foresight to shift their center of gravity before the moment of no return.