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.

Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 30

Chapter 3, Problem 30

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.

Verified step by step guidance

Step 1: Understand that displacement is the vector sum of the individual displacements during each segment of motion. Displacement for each segment can be calculated using the formula:

d = v ⋅ t

, where

v

is the velocity vector and

t

is the time duration.

Step 2: Calculate the displacement for the first segment of motion. Use the velocity vector

v

₁ = (55î − 10ĵ) mph and the time

t = 1.0

h. Multiply the velocity vector by the time:

d

₁ = (55î − 10ĵ) ⋅ 1.0.

Step 3: Calculate the displacement for the second segment of motion. Use the velocity vector

v

₂ = (20î + 50ĵ) mph and the time

t = 2.0

h. Multiply the velocity vector by the time:

d

₂ = (20î + 50ĵ) ⋅ 2.0.

Step 4: Add the two displacement vectors to find the total displacement. Use the formula:

d

_total = d₁ + d₂. Substitute the expressions for

d

₁ and

d

₂ from Steps 2 and 3.

Step 5: Write the total displacement in component form using unit vectors. Combine the

î

and

ĵ

components separately to express the result as

d

_total = (xî + yĵ), where

x

and

y

are the combined components.

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Key Concepts

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

Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude and direction, which means it can be represented in component form using unit vectors. In this question, Trevon's velocities v₁ and v₂ are given in terms of unit vectors, indicating how far he travels in the x (î) and y (ĵ) directions.

Displacement

Displacement is the vector that represents the change in position of an object. It is calculated as the final position minus the initial position and is also expressed in component form. In this scenario, Trevon's total displacement will be the sum of the displacements from each segment of his journey, taking into account the time spent traveling at each velocity.

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 this problem, Trevon's total displacement is found by calculating the displacement for each leg of his journey and then adding these vectors together to find the overall displacement in component form.

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Vector Addition By Components

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