The Inertial Frame may be referred to as the "ground".  The Non-Inertial Frame includes the MGR and the two riders.

The problem difficulty is indicated by the number of stars.

*1.  Click on Initialize/Rest to set the parameters to their original values.  Toggle to the Inertial Frame if necessary.

1. Determine by calculation what Vo must be (at q = 0°) so that the ball crosses the MGR's path at (x,y) = (0,R).  Don't test your result yet.
2. When the ball is at (x,y) = (0,R), where will the Blue rider be?  Give Blue's (x,y) coordinates.
3. Now test your predictions.  Don't forget to Reset.  You can use the Step buttons to position the ball as close to (0,R) as possible.

*2.  Your problem now is to throw the ball so that it passes through the center of the MGR.  This will require a negative q, of course, but you must determine exactly what the angle is.

1. Set Vo to 20 m/s.  Then calculate what q must be.  Test your prediction.
2. Now double Vo.  What must q be?  Test your prediction.

**3.  In problem 2, you probably used the fact that the y-component of the ball's velocity relative to Blue had to be the opposite of the y-component of Blue's velocity relative to the ground at the instant the ball was thrown.  In this way, the y-component of the ball's velocity relative to the ground would be 0.  That is, at t = 0,

Vy,ball wrt ground = Vy,ball wrt Blue + Vy,Blue wrt ground       ("wrt" = "with respect to")

0 = Vy,ball wrt Blue + Vy,Blue wrt ground

Vy,ball wrt Blue = -Vy,Blue wrt ground

1. Now we'll make the problem a bit more difficult.  Not only must the ball pass through the center of the MGR, it must also be caught by Blue.  That is, the ball and Blue must reach (x,y) = (-R,0) at exactly the same time.  That means you'll need to use another equation, this one involving time.  Determine now what Vo and q must be given T = 5 s.  Test your prediction.
2. Toggle to the Non-Inertial Frame and run the applet again.  We think you'll find the path interesting.
3. Change T to 2.5 seconds and toggle back to the Inertial Frame.  What must Vo and q be now in order for Blue to catch the ball?  (We hope you found in part a that q is independent of the MGR's speed.  Vo, however, will change.)  Test your prediction.

**4.  With T = 2.5 seconds, what must Vo and q   be so that Red catches the ball after one full rotation?  Test your prediction.  Look at it from the Non-Inertial Frame, too.  How do Vo and q depend on T?

**5.  With T = 5 seconds, what must Vo and q  be so that Red catches the ball after one quarter rotation?  (The ball cannot pass through the center in this case.)

***6.  Suppose Red is to catch the ball after rotating through any angle a less than p/2.

1. Determine formulas for Vo(a) and q(a).  Then check them for particular values of a.
2. In a real situation on the MGR where one rider wants to throw the ball to the other, a will generally be small.  Using the small angle approximations, sina = a and cosa = 1, eliminate a from the equations from part a in order to obtain Vo as a function of q only.  Then use the applet to investigate the range of angles, q, for which the approximation works well.