Question
5. A boat is being pulled toward a dock by a rope attached to its bow through a pulley on the dock. The pulley is higher than the boat's bow by several feet. Relate the rate that the rope is hauled in with how fast the boat is approaching the dock.
Answer
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Emilia
Elite · Tutor for 8 years
Answer
The rate at which the rope is hauled in is higher than the speed at which the boat is approaching the dock. This is due to the fact that the boat does not move in a straight line towards the dock given the triangular setup (from the dock, boat, and the elevated position where the rope is located), and the boat is being pulled upward in accordance to the height to which the pulley is installed above the dock. The actual speed that the boat nears the dock depends on the horizontal component of the movement, not the total length the rope is brought in as can be misunderstood to.
Explanation
This question is asking us to establish a relationship between the rate at which the rope is hauled in, and how fast the boat approaches the dock.To answer this question accurately, one must first gain familiarity with the concept of rates and trigonometry within the context of physics. Relative speeds are, in essence, represented here. This problem is a good demonstration of rates in every day situations.We must visualize a right triangle described here formed by the boat, dock and rope. Normally, as the correct physical understanding would insist, one would tend to think that the speed at which the rope is pulled would simply be the speed or rate at which the boat approaches the dock. However, do keep in mind that there exist angles in the invisible triangle formed between the pulley, boat and level of the water of the dock. Since the rope is being pulled using a pulley set above on a dock, this makes it so that the boat cannot move directly toward the bow, or horizontally straight to it. Instead, the movement will be slant, or diagonal in approach. Due to the existence of such an angle thus, although the rope is tugged in 'vertically straight', the boat's actual movement towards the dock shows difference due to the angle instigated by the setup. This is basic principle of a triangle's properties.This problem presents an excellent context in relation to the idea of vectors within physics. To calculate speeds or movements involving angles, components of vectors, such as a component moving horizontally, a component moving vertically etc are often considered.Hence, we may infer that actual realization of the boat's movement is the horizontal component taken from the speed or rate at which the rope was hauled in.