Tossing a Ball on a Carosel.
If I pass a playground and it has a merry-go-round (MGR), I can't resist riding on it and playing with the dynamics of rotating systems. I especially like tossing stones across the MGR and watching the path deviate from a straight line one would see in a non-rotating system. For this reason I wanted to do the numbers on a realistic ball toss on a larger rotatiing system , for example, a carosel.
1. Establish the size and speed.
- Radius of Carosel = 5 meters
- Period of Rotation = 10 seconds
The time for the ball to fly across 10 m rim to rim when the Carosel is not turning is 1.26 seconds and the speed of the toss must be 10.1 m/s at an angle of 39 degrees above the horizontal.
When the Carosel is rotation an additional component of the toss velocity is added relative to the fixed frame of reference ( The one of an observer not on the Carosel). That is the speed of the tosser on the rim moving at 3.12 m/s.
This causes the ball to take a trajectory at an angle of 21.6 degrees from direction to the center of the carosel. This is shown in the diagram of the moving carosel from above. When the ball leaves the tosser's hand it will travel as shown.
In this diagram, the carosel is rotating in a counter-clockwise direction. The ball will reach the rim in 1.09 seconds which is only 9.3 metres along the path.
In order for someone to catch the ball, they would have to be 39.2 degrees away from where the ball crosses the rim at the time of the toss. That is 97 degrees away from the tosser.
In this diagram the dynamics of the tossed ball is relatively easy to calculte.
If we were to take the view of the person throwing and the person catching the ball. It would be quite a bit more compliationed and with not such simple geometry.
The two animations given below illustrates this. In this case the carosel is rotating clockwise and show the path of the ball and the tosser. The left animation is what an outside observer would see. The animation on the right shows what is seen in the rotating frame.
