Question

A hiker travels \(1.50 \mathrm{~km}\) north and turns to a heading of \(20.0^{\circ}\) north of west, traveling another \(1.50 \mathrm{~km}\) along that heading. Subsequently, he then turns north again and travels another \(1.50 \mathrm{~km} .\) How far is he from his original point of departure, and what is the heading relative to that initial point?

Short answer

Answer: The hiker's total displacement vector has a magnitude of 4.45 km and a direction of 276.6° North of West.

Step-by-step solution

Break down each leg of the journey into x and y components

The hiker travels three legs during his journey: 1. First leg: 1.50 km North. This leg only has a vertical (y) component since it's purely in the north direction. - x1 = 0 km - y1 = 1.50 km 2. Second leg: 1.50 km at 20.0° North of West. We need to find the x and y components using trigonometry. - x2 = 1.50 km * sin(20.0°) = 1.50 km * 0.3420 = 0.5130 km (West direction) - y2 = 1.50 km * cos(20.0°) = 1.50 km * 0.9397 = 1.4096 km (North direction) 3. Third leg: 1.50 km North. This leg also only has a vertical (y) component, similar to the first leg. - x3 = 0 km - y3 = 1.50 km

Add the x and y components to find the total displacement

Add the individual components of the three legs to find the total displacement: - Total x displacement: x1 + x2 + x3 = 0 + (-0.5130) + 0 = -0.5130 km (West direction) - Total y displacement: y1 + y2 + y3 = 1.50 + 1.4096 + 1.50 = 4.4096 km (North direction)

Calculate the magnitude of the total displacement

Using the Pythagorean theorem, find the magnitude of the total displacement: Total displacement = \(\sqrt{(-0.5130 \mathrm{~km})^2 + (4.4096 \mathrm{~km})^2} = 4.45 \mathrm{~km}\).

Calculate the direction of the total displacement

To find the direction, get the angle using the arctangent function: Angle (\(\theta\)) = \(\arctan{\frac{y}{x}} = \arctan{\frac{4.4096 \mathrm{~km}}{-0.5130 \mathrm{~km}}}\) \(\theta \approx -83.4^{\circ}\) (measured clockwise from North) Since we are looking for a heading relative to the initial point, we change to counterclockwise from North: Relative heading = \(360^{\circ} - 83.4^{\circ} = 276.6^{\circ}\) North of West. The hiker is 4.45 km away from his original point of departure, with a heading of 276.6° North of West.

Key concepts

Trigonometry in Physics

Understanding the role of trigonometry in physics is essential for tackling problems involving angles and directional movement. In the context of vector displacement, trigonometry allows us to break down a movement into its horizontal (x) and vertical (y) components. This is particularly helpful when dealing with motion at an angle to the standard axes.

For example, if a hiker travels at a certain angle of elevation or depression, we can use the sine, cosine, and tangent functions to determine the respective x and y components of their displacement. Specifically, the sine function is used to calculate the x component (horizontal) when the vector is at an angle to the vertical axis, while the cosine function calculates the y component (vertical). The tangent function can be particularly useful for finding the angle of direction if we know the components. This process is the key to solving complex motion problems in a two-dimensional plane.

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. In the exercise involving the hiker, the displacement vector for each leg of the journey was broken down into its x (horizontal) and y (vertical) components. By calculating each component, we can understand how much of the hiker's movement was in the North-South direction (y-axis) and how much was in the East-West direction (x-axis).

When a vector is at an angle, its components can be determined using trigonometric functions. For instance, in the second leg of the hiker's journey, the displacement vector at a 20° angle to the West required the use of the sine function to find the x component and the cosine function to calculate the y component. By finding these, we can effectively simplify the analysis of each leg of the journey and be prepared to combine these components to find the overall vector displacement.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone concept within the study of vectors in physics. It states that in a right-angled triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the lengths of the other two sides. In the context of vectors, when we break down a vector into its orthogonal components (x and y), these components form the legs of a right triangle, and the vector itself represents the hypotenuse.

To find the magnitude of the resulting vector after the hiker has completed all parts of the journey, we use the Pythagorean theorem. By squaring each component, adding them together, and then taking the square root, we arrive at the total distance the hiker is from the starting point. This theorem is what allows us to move from the individual movements along the x and y axes to the overall displacement vector, which is a fundamental step in solving such physics problems.

Magnitude and Direction of Vectors

Every vector in physics has two fundamental properties: magnitude and direction. The magnitude refers to the size or length of the vector, while the direction indicates the vector's orientation. After determining the components of a vector, we then aim to find these two properties to describe the vector fully.

The magnitude of the total displacement vector is its length and can be calculated using the Pythagorean theorem, as seen with the hiker's journey. To find the direction, we can use trigonometry, specifically the arctan function, which yields the angle of the vector relative to the horizontal axis. In the case of the exercise, this provided the heading of the hiker relative to the original starting point. Direction is often represented by an angle, and it's vital to consider which way the angle is measured, typically either from the North direction (in geographical contexts) or from the positive x-axis (in standard Cartesian coordinates).