I guess you are in the southern hemisphere, because you are talking about the Tropic of Capricorn? That is at about 22 south latitude. I get the equator speed is about 464 m/s. To get half that, or 232 m/s, the catcher would have to be at 60 south, not 22.
There is a way to simplify the problem.
Make the earth into a flat disk with the south pole at the center (and the north pole underneath it.
If you (the thrower) are squarely facing the catcher, that means the catcher and you are moving around each other, because you are both on the earth which is turning.
You can throw directly at the catcher, which is directly at the south pole.
Here's the picture from the viewpoint of space.
You see, it depends on how fast you throw it, compared to how fast the earth is turning.
If you throw really really fast, the ball will get to the catcher immediately.
If you throw it less fast, but still fast enough to get to the catcher's latitude in one time unit, you see the catcher will have travelled less distance in that time, so it will look as if the ball has curved to the East.
If you throw even less fast, you can see that the catcher will see it curve to the East, and may even keep turning and go away.
If you throw it really slowly, it will just curve away from both of you and drift off into space, appearing to make cycles just because you are making cycles.
Of course, it gets more complicated when gravity gets involved, which at low speeds keeps thing from drifting off into space.
In that case, the ball will basically go into an elliptical orbit, which from your viewpoints will look like crazy cycles (assuming it isn't caught, there is no air and no ground to crash into).