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 29a

Chapter 3, Problem 29a

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?

Verified step by step guidance

Step 1: Identify the vertical components of the morning and afternoon displacements. From the problem, the vertical component of the morning displacement is 200 m (upward), and the vertical component of the afternoon displacement is -300 m (downward).

Step 2: Add the vertical components of the morning and afternoon displacements to determine the net vertical displacement. Use the equation: Sᵥₑᵣₜᵢ𝚌ₐₗ = Sₘₒᵣₙᵢₙ₉ᵥₑᵣₜᵢ𝚌ₐₗ + Sₐբₜₑᵣₙₒₒₙᵥₑᵣₜᵢ𝚌ₐₗ.

Step 3: Substitute the values into the equation: Sᵥₑᵣₜᵢ𝚌ₐₗ = 200 m + (-300 m).

Step 4: Simplify the expression to find the net vertical displacement. This will tell you how much higher or lower you are compared to your starting point.

Step 5: Interpret the result. If the net vertical displacement is positive, you are higher than your starting point. If it is negative, you are lower than your starting point.

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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 in the east-west, north-south, and vertical directions, 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 in each direction. In the given problem, the total displacement for both the morning and afternoon hikes must be calculated by summing their respective components to find the overall change in position.

Vertical Displacement

Vertical displacement refers specifically to the change in height of an object relative to a reference point, typically the ground. It is crucial for determining how much higher or lower one is compared to the starting point. In this scenario, calculating the vertical displacement involves summing the vertical components from both the morning and afternoon hikes to assess the overall elevation change.

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Find a vector that points in the same direction as the vector ( î ＋ ĵ ) and whose magnitude is 1.

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Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward's displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

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Textbook Question

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?

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Textbook Question

Trevon drives with velocity v₁ = (55î - 10ĵ) mph for 1.0 h, then v₂ = (20î ＋ 50ĵ) mph for 2.0 h. What is Trevon's displacement? Write your answer in component form using unit vectors.

Textbook Question

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.

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Textbook Question

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