Circular Motion and Projectiles
Math 211  Prof. Leise

Email completed labs to tleise@amherst.edu by noon next Wednesday.

Today's lab will give you practice with vector functions describing circular motion and parabolic trajectories of projectiles.

Clear[t,t0,t1] (* Hit shift-enter to evaluate this first.  If you get error messages during your work, one thing to try is to go back to this line and hit shift-enter again. *)

You want to toss a disposable camera to a friend who is riding a small Ferris wheel at the county fair (he forgot to bring it with him and he wants to take a picture at the top).

The Ferris wheel rotates counterclockwise once every 100 seconds, with the axis 16 meters above the ground and seats 15 meters from the axis.

If your friend is at the bottom at time t=0, find a vector-valued function that describes his position (FWx,FWy) as the wheel rotates.  Fill in expressions for FWx and FWy, then hit shift-enter on each:

FWx=

FWy=

You stand 25 meters from the base of the Ferris wheel (to the right of the wheel) and you throw the camera when your hand is a meter above the ground with speed 20 meters per second at an angle of 60 degrees from horizontal.  Find a vector-valued function that describes the motion of the camera if you throw the camera at time t=t0 (you have to wait a while to allow your friend to rotate closer to you).

Enter the x and y coordinate functions for the camera's path.  Remember that it is thrown at time t0, so these functions will involve the time since being thrown, (t-t0), rather than just t.  Fill in expressions for Cx and Cy, then hit shift-enter:

Cx=

Cy=

Solve for the time t0 (when camera should be thrown) and the time t when the camera will be caught by using the FindRoot command: FindRoot[{Cx==FWx,Cy==FWy},{{t0,guess},{t,guess}}], where you need to enter guesses for the times. If the answer is negative or seems unreasonable, you need to enter better guesses (there are many solutions, since you  could wait for the Ferris wheel to rotate several times before throwing the camera, but we want the solution with minimum wait). Then enter the correct times in the cell below (and hit shift-enter).

t0=    (* time camera is thrown *)

t1=    (* time camera is caught *)

If your Ferris wheel and camera position functions are correct (and you hit shift-enter on the three pairs of values above), then the code below should draw the path of your friend in the wheel and the camera once it is thrown.

FerrisWheel=ParametricPlot[{FWx,FWy},{t,0,t1},PlotStyle->Blue];
Camera=ParametricPlot[{Cx,Cy},{t,t0,t1},PlotStyle->Red];
Show[FerrisWheel,Camera,AspectRatio->Automatic,PlotRange->All,PlotLabel->"Ferris Wheel in Blue, Camera Path in Red"]