Problem 49 - Fundamentals of Physics

Question

A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side, \(90.0 \mathrm{~km}\) due north. The sailor, however, ends up \(50.0 \mathrm{~km}\) due east of the starting point. (a) How far and (b) in what direction must the sailor now sail to reach the original destination?

Step-by-Step Solution

Answer

The sailor must travel 103.0 km at 29.1° east of north to reach the destination.

Step 1: Conceptualize the Setup

To solve the problem, consider the initial path and the actual ending point. The original plan was to travel 90 km due north. Instead, the sailor ended up 50 km east of the starting point. These can be visualized as two sides of a right triangle.

Step 2: Determine the Resultant Distance

To find out how far the sailor must now travel, calculate the hypotenuse of the right triangle formed with the north and east components of 90 km and 50 km, respectively. Use the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} \]where \(a = 90\) km, \(b = 50\) km.Calculate:\[ c = \sqrt{90^2 + 50^2} = \sqrt{8100 + 2500} = \sqrt{10600} \approx 103.0 \text{ km} \]

Step 3: Calculate the Direction of Travel

The direction can be calculated using the tangent of the angle, which is the ratio of the side opposite the angle to the side adjacent (east component to north component):\[ \theta = \tan^{-1} \left( \frac{50}{90} \right) \]Calculate:\[ \theta = \tan^{-1} (0.5556) \approx 29.1^\circ \]This angle is measured from the north direction towards the east.

Step 4: Formulate the Final Answer

The sailor must travel approximately 103.0 km at an angle of 29.1° east of north to reach the original destination.

Key Concepts

right triangle

A right triangle is a triangle in which one of the angles is precisely 90 degrees. This is an extremely useful shape in geometry, particularly for calculating distances and angles in two-dimensional space. In the context of this exercise, imagining the journey of the sailor forms a right triangle on the map, with the northern leg being 90 km and the eastern leg being 50 km. The right angle is formed where these two directions meet, which allows us to utilize the Pythagorean theorem efficiently.

• The horizontal leg of the triangle (eastward journey) is 50 km.
• The vertical leg of the triangle (the intended northward journey) is 90 km.
• The hypotenuse represents the shortest path the sailor must now take.

Recognizing the shape and orientation of a right triangle in real-world contexts can simplify complex navigational problems, like determining the shortest path across a large lake.

distance calculation

Distance calculation often involves using the Pythagorean theorem, especially when dealing with right triangles. This theorem provides a way to calculate the length of the hypotenuse when the lengths of the other two sides are known.
For this sailboat problem, calculate the distance directly using the equation: \[ c = \sqrt{a^2 + b^2} \]where:

• \( a \) is the length of one side of the right triangle (90 km north in this problem).
• \( b \) is the length of the other side (50 km east).
• \( c \) will be the hypotenuse, indicating the direct distance needed to reach the planned destination.

Using the values, calculate:\[ c = \sqrt{90^2 + 50^2} = \sqrt{8100 + 2500} = \sqrt{10600} \approx 103.0 \text{ km} \]Thus, the sailor must travel approximately 103.0 km from the current position to reach the destination directly.

angle of direction

Understanding the angle of direction is crucial for navigational tasks. The angle helps determine the exact path to head towards a specific destination. For calculating direction, trigonometry comes in handy. Specifically, the tangent function relates the two perpendicular sides of a right triangle.
In this scenario, calculate the angle from north (our reference direction) using the tangent of the angle where:

• The opposite side is the eastward distance (50 km).
• The adjacent side is the northward distance (90 km).

This angle \( \theta \) can be found using:\[ \theta = \tan^{-1} \left( \frac{50}{90} \right) \approx 29.1^\circ \]This angle represents the direction the sailor must turn from north, heading a bit towards east this time, to successfully arrive at the original target location. Knowing this precise angle allows the sailor to adjust their course efficiently.