The boat is pulled to the shore


A boy stands on a shore and pulls a rope connected to a boat. The velocity of the rope vr is constant. Prove that the closer the boat to the shore, the faster it moves.


We can start with the Pythagorean formula.

h2 + s2 = l2

The length of the rope l and the distance between the boat and shore s are functions of time. The height from the sea level to the boy’s hand is constant.

We differentiate the Pythagorean expression with respect to time and we get

d(h2)/dt + d(s2)/dt = d(l2)/dt


2s*ds/dt = 2l*dl/dt

Now, we can use the fact that s/l = cos(α) , where α depends on time.
Also, it has to be clear that

ds/dt = vb, where vb is the velocity of the boat and

dl/dt = vr.

Finally, we got the following expression

vb = vr/cos(α).

From the last expression, we can see that when the boat approaches the shore it moves faster, because cos(α) decreases as α increases, so vb increases as the velocity of the rope vr remains constant.


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