Why does the boat move faster than the rope when being pulled to shore?

[SprucerMoose]

G'day everyone,

I was not sure where to exactly post this question, but here goes.

I am currently doing some calculus homework and although I have found the solution to the problem, I have discovered something I have never seen before.

The example pertains to a boat being pulled to shore by a rope, which runs through an elevated ring on shore. The person pulling the rope is somewhere on shore at ground level. If the location of the boat is point A, the ring point B and the ground immediately below the ring is point C, triangle ABC is right angled.

I calculated from pythagoras theorem that D'(t) = (L(t)L'(t))/D(t), where L is the length of rope from AB at a given time and D is the distance AC. This shows that as long as the ring is elevated, owing to L(t) always being larger than D(t), D'(t) is always larger than L'(t) and the boat always moves to shore faster than the speed at which the rope is pulled.

This is tripping me out for some reason. I cannot picture how this occurs and was wondering if someone could break it down a little so I can get an intuitive understanding of this senario.

Thanks

 

[Daever]

just picture this way: the total distance of rope used will be the length of the hypotenuse(AB) minus the opposite(BC), this will allways be smaller than the adjacent length(AC). You have to cover both distances at the same time so the velocity has to be greater along the adjacent length because its a larger distance than what the rope has to travel. Now, think of point B being really high; now being really low - even flat. Visualise what happens between these two extrema when B is moved up and down, you'll notice the rope, even though longer has to travel a shorter distance the higher you go, making the velocity of the boat increase!