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An object moves along a line where its velocity function is as described by the graph. What is the total distance that it has traveled between [2, 7]? VCD F 10 v(t)

An object moves along a line where its velocity function is as described by the graph. What is the total distance that it has traveled between [2, 7]? VCD F 10 v(t)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Velocity and Distance Problem #### Problem Description: An object moves along a line where its velocity function is as described by the graph. The question posed is: What is the total distance that it has traveled between time \( t = 2 \) and \( t = 7 \)? #### Graph Details: - **X-axis (Horizontal Axis):** Represents time (\( t \)), ranging from 0 to 10. - **Y-axis (Vertical Axis):** Represents velocity (\( v(t) \)), with values ranging from -3 to 3. The velocity function \( v(t) \) changes over time as follows: - From \( t = 0 \) to \( t = 2 \), the velocity is constant at -2. - At \( t = 2 \), the velocity starts increasing linearly reaching 0 at \( t = 3 \). - From \( t = 3 \) to \( t = 5 \), the velocity increases further linearly, reaching 2 at \( t = 5 \). - From \( t = 5 \) to \( t = 6 \), the velocity stays constant at 2. - At \( t = 6 \), the velocity decreases to 1. - From \( t = 7 \) onward, the velocity remains constant at 1. #### Multiple Choice Answers: - \( \bigcirc \) 2 - \( \bigcirc \) 4 - \( \bigcirc \) 6 - \( \bigcirc \) 8 #### Analysis: To determine the total distance traveled by the object between \( t = 2 \) and \( t = 7 \), we'll calculate the area under the velocity curve for each segment. 1. **From \( t = 2 \) to \( t = 3 \):** - The velocity increases linearly from -1 to 0. - This forms a right triangle with base 1 (time) and height 1 (velocity). - Area (distance) = \( \frac{1}{2} \times 1 \times 1 = 0.5 \) 2. **From \( t = 3 \) to \( t = 5 \):** - The velocity increases linearly from 0 to 2. - This forms a trapezoid with bases 0 and 2, and height of
Transcribed Image Text:### Velocity and Distance Problem #### Problem Description: An object moves along a line where its velocity function is as described by the graph. The question posed is: What is the total distance that it has traveled between time \( t = 2 \) and \( t = 7 \)? #### Graph Details: - **X-axis (Horizontal Axis):** Represents time (\( t \)), ranging from 0 to 10. - **Y-axis (Vertical Axis):** Represents velocity (\( v(t) \)), with values ranging from -3 to 3. The velocity function \( v(t) \) changes over time as follows: - From \( t = 0 \) to \( t = 2 \), the velocity is constant at -2. - At \( t = 2 \), the velocity starts increasing linearly reaching 0 at \( t = 3 \). - From \( t = 3 \) to \( t = 5 \), the velocity increases further linearly, reaching 2 at \( t = 5 \). - From \( t = 5 \) to \( t = 6 \), the velocity stays constant at 2. - At \( t = 6 \), the velocity decreases to 1. - From \( t = 7 \) onward, the velocity remains constant at 1. #### Multiple Choice Answers: - \( \bigcirc \) 2 - \( \bigcirc \) 4 - \( \bigcirc \) 6 - \( \bigcirc \) 8 #### Analysis: To determine the total distance traveled by the object between \( t = 2 \) and \( t = 7 \), we'll calculate the area under the velocity curve for each segment. 1. **From \( t = 2 \) to \( t = 3 \):** - The velocity increases linearly from -1 to 0. - This forms a right triangle with base 1 (time) and height 1 (velocity). - Area (distance) = \( \frac{1}{2} \times 1 \times 1 = 0.5 \) 2. **From \( t = 3 \) to \( t = 5 \):** - The velocity increases linearly from 0 to 2. - This forms a trapezoid with bases 0 and 2, and height of
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