(2-3)  Let’s see if you can estimate how fast the merry-go-round must go in order to make Nefertiti’s dream come true. First define some terms:

Distance each sardine must fly to hit you = D

velocity of throw (relative to Nefertiti) = v             

velocity of Nefertiti on merry-go-round (relative to you)  = u

(in the direction of the tangent )

                    time for 1 rotation = T            radius of merry-go-round = r

We’ll assume that the release points for the sardines are exactly a half turn apart, and that Nefertiti’s throw is aligned with her velocity vector on the merry-go-round at the point of release, as shown in the picture. In reality this may not be quite true but so long as you stand far away, relative to the radius of the merry-go-round, it only introduces a small error, as Pythagoras will confirm, if you ask him.

①  First write down expressions for the total velocities of sardines 1 and 2.

②  Deduce expressions for t(1) and t(2), defined thus: 

t(1) = time from release of sardine 1 to the moment it hits you.

t(2) = time from release of sardine 1 to the moment sardine 2 hits you.

③  For the sardines to hit simultaneously, we require t(1) = t(2).  So now equate your two expressions and do a bit of algebra to get an expression for T, the rotation time of the merry-go-round.

④  Now let’s focus on Nefertiti and the merry-go-round.  Our aim, remember, is to figure out how fast we need it to rotate to achieve the simultaneous sardine impact, which we can express as the time for one rotation, T. Nefertiti completes one tour of the circumference in time T, so her rotational speed is given by circumference / T. Her velocity vector is always directed along the tangent to the merry-go-round, at whatever point in the rotation she is.  So:

·         Deduce an expression for her velocity in terms of T

·         Substitute this into the expression you derived in ③.

·         Do a bit of gentle rearranging to derive an expression for the value of T that Nefertiti needs.

⑤  We can put in some numbers to see how this works. Let’s say the merry-go-round radius is 1m, Nefertiti’s throw speed (without the help or hindrance of her rotation) is 15 ms-1 and you are standing 20m away from her when she lets fly. Calculate the rotation time, T, that Nefertiti requires.