Man pulling a boat along the water with a rope (from above on a steep bank)

[kuruman]

"Mankind is composed of two kinds of people - those who think Mankind is composed of two kinds of people, those who don’t think Mankind is composed of two kinds of people, and those who cannot count."

I've heard that Mankind is composed of 10 kinds of people, those who understand binary numbers and those who don't.

 

[kuruman]

I'd like @haruspex or someone else to show me where am going wrong.

Here is another way to look at the trajectory in space of a point $Q$ on the rope at distance $PQ$ from the boat along the rope. The key features to understand is that the rope always points towards the hands of the man at point O. This means that the trajectory of $Q$ in space is independent of whether the rope is pulled at a constant rate or whether the boat moves at constant speed. Point $Q$ will move along a uniquely defined path as angle $\alpha$ increases to $\frac{\pi}{2}.$

The figure on the right shows segment $PQ$ (in red) at two successive positions of the boat. In time interval $dt$ the boat advances to the right by amount $~dx=u(t)dt~$ and segment $PQ$ simultaneously rotates counterclockwise about point $P$ by an arc of length $~ds=(PQ)d\theta$. Note that $d\theta=-d\alpha$.

You can see from the drawing that
$d\mathbf x=dx~\mathbf{\hat x}$ and that
$d\mathbf s=-ds\sin\!\alpha~\mathbf{\hat x}+ds\cos\!\alpha ~\mathbf{\hat y}$ (Angle $\alpha$ is not shown to avoid clutter.) With $~ds=(PQ)~d\alpha$, the displacement of point $Q$ in time $dt$ is $$d\mathbf r=\left[dx-(PQ)\cos\alpha~d\alpha\right]~\mathbf{\hat x}+(PQ)~\sin\alpha~d\alpha~\mathbf{\hat y}.$$From this you can see that all points along the rope move simultaneously

• to the right by a the same amount $dx$
• to the left by an amount that is proportional to how far away ($PQ$) they are from the boat
• up by an amount that is also proportional to how far away they are from the boat.

(Edited to fix equation typos.)