Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 29b

Chapter 3, Problem 29b

While vacationing in the mountains you do some hiking. In the morning, your displacement is $$\mathbf{S}$_{morning} = (2000 \, $\text{m}$, $\text{east}$) + (3000 \, $\text{m}$, $\text{north}$) + (200 \, $\text{m}$, $\text{vertical}$)$ . Continuing on after lunch, your displacement is $$\mathbf{S}$_{afternoon} = (1500 \, $\text{m}$, $\text{west}$) + (2000 \, $\text{m}$, $\text{north}$) - (300 \, $\text{m}$, $\text{vertical}$)$ . What is the magnitude of your net displacement for the day?

Verified step by step guidance

Step 1: Break down the given displacements into their respective components. For the morning displacement (Sₘₒᵣₙᵢₙ₉), the components are: east = 2000 m, north = 3000 m, vertical = 200 m. For the afternoon displacement (Sₐբₜₑᵣₙₒₒₙ), the components are: west = -1500 m (negative because west is opposite to east), north = 2000 m, vertical = -300 m (negative because it is downward).

Step 2: Add the components of the morning and afternoon displacements to find the net displacement components. For the east-west direction: 2000 m (east) + (-1500 m) (west). For the north direction: 3000 m (north) + 2000 m (north). For the vertical direction: 200 m (upward) + (-300 m) (downward).

Step 3: Combine the net components into a single vector. The net displacement vector will have components in the form: (net east-west, net north, net vertical).

Step 4: Calculate the magnitude of the net displacement vector using the formula for the magnitude of a vector:

\sqrt{x^{2} + y^{2} + z^{2}}

, where x, y, and z are the net components in the east-west, north, and vertical directions, respectively.

Step 5: Substitute the values of the net components into the formula and simplify to find the magnitude of the net displacement. Ensure all units are consistent (meters in this case).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Displacement

Displacement is a vector quantity that refers to the change in position of an object. It is defined as the shortest distance from the initial to the final position, taking into account the direction. In this context, displacement is expressed in terms of its components along the east-west, north-south, and vertical axes, allowing for a comprehensive understanding of the overall movement.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. This involves adding the corresponding components of the vectors, which can be visualized graphically or calculated mathematically. In the given problem, the total displacement for the day is found by adding the morning and afternoon displacement vectors component-wise.

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size, regardless of its direction. It can be calculated using the Pythagorean theorem for vectors in two or three dimensions. In this scenario, after determining the net displacement vector from the sum of the morning and afternoon displacements, the magnitude will provide the total distance traveled in a straight line from the starting point.

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The minute hand on a watch is 2.0 cm in length. What is the displacement vector of the tip of the minute hand in each case? Use a coordinate system in which the y-axis points toward the 12 on the watch face. From 8:00 to 8:20 a.m.

While vacationing in the mountains you do some hiking. In the morning, your displacement is $S_{m o r n i n g} = (2000 m, east) + (3000 m, north) + (200 m, vertical)$ . Continuing on after lunch, your displacement is $S_{a f t e r n o o n} = (1500 m, west) + (2000 m, north) - (300 m, vertical)$ . At the end of the hike, how much higher or lower are you compared to your starting point?

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