Some children are practicing catching baseballs. The coach tosses the ball into the air and hits the ball with his bat. The ball travels nearly horizontally, directly at the short stop who manages to catch the line drive. Did the coach, via the bat, do any work on the ball as it was hit

Yes he did

Explanation:

Horizontally speaking, the ball didn't have any speed prior to the bat hit by the coach. After the hit, the ball has horizontal speed, or kinetic energy. This energy must come from the work performed by the coach during the hit. This work is the product of the hit force and and distance it travels during the brief time the ball makes contact with the bat.

Work is defined as the amount of movement done by a force, and it can be calculated as:

W = f*d

where W is the work, f is the force and d is the distance.

So when the coach hits the ball, he is appliying some force in the ball (via the bat) that impulses the ball briefly (we can considerate that the force is only applied in a differential of time dt)

But we have a force applied and a displacement of the ball, so yes, the bat does work on the ball.