## Relative Motion on a Merry-Go-Round

Level of difficulty:  variable, algebra to precalculus

In these problems, two people ride on a merry-go-round (radius fixed at 10 m..it's big!).  One rider tosses a ball toward the other.   The period of motion can be varied as can the initial velocity of the ball, relative to rider.  One of the problems is to select the initial velocity of the ball so that it passes through center of the merry-go-round.  Another is for the rider who throws the ball to catch it him/herself after exactly half a rotation.

A unique feature of the problem allows the user to toggle between inertial and non-inertial reference frames.  The animated gif to the right demonstrates the view from, say, an overhead camera (inertial frame).  The non-inertial point of view is that of a camera that rotates at the same frequency as the merry-go-round.   Vertical motion of the ball is ignored.  One could pretend, for example, that the riders, strapped to a giant, horizontal disc, slide a puck across the disc's frictionless surface.

The output of one problem, as seen from the inertial frame, is shown below.  The blue rider threw the ball (green) with an initial velocity such that the same rider caught the ball after half a rotation.  The ball actually had to be thrown at about a 57° angle away from the direction of rotation.  This is seen from the non-inertial point of view, also shown below.  (The animations below are not necessarily synchronized.)

Inertial viewpoint

Non-inertial viewpoint

These problems may be used in conjunction with the Physics Merry-go-Round lab.