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.