For 0 st≤ 7, a particle moves along the x-axis. The velocity of the particle is given by v(t) = sin(nt/3). The particle is at position x = -5 when t = 0. (a) For 0 st≤ 7, when is the particle moving to the right? (Enter your answer using interval notation.) Justify your answer. O At these times, x(t) < 0. At these times, v(t) > 0. O At these times, v(t) < 0. O At these times, a(t) < 0. O At these times, x(t) > 0. O At these times, a(t) > 0. (b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time t = 0 to t = 7. 1) dt (c) Find the acceleration of the particle at time t = 2. (Round your answer to two decimal places.)
For 0 st≤ 7, a particle moves along the x-axis. The velocity of the particle is given by v(t) = sin(nt/3). The particle is at position x = -5 when t = 0. (a) For 0 st≤ 7, when is the particle moving to the right? (Enter your answer using interval notation.) Justify your answer. O At these times, x(t) < 0. At these times, v(t) > 0. O At these times, v(t) < 0. O At these times, a(t) < 0. O At these times, x(t) > 0. O At these times, a(t) > 0. (b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time t = 0 to t = 7. 1) dt (c) Find the acceleration of the particle at time t = 2. (Round your answer to two decimal places.)
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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![For educational purposes, here is the transcription and explanation of the image content:
**Problem Statement:**
Do not use technology.
For \( 0 \leq t \leq 7 \), a particle moves along the x-axis. The velocity of the particle is given by \( v(t) = \sin(\pi t / 3) \). The particle is at position \( x = -5 \) when \( t = 0 \).
**(a)** For \( 0 \leq t \leq 7 \), when is the particle moving to the right? (Enter your answer using interval notation.)
[Blank box for response]
**Justify your answer:**
- ○ At these times, \( x(t) < 0 \).
- ● At these times, \( v(t) > 0 \). (Selected)
- ○ At these times, \( v(t) < 0 \).
- ○ At these times, \( a(t) < 0 \).
- ○ At these times, \( x(t) > 0 \).
- ✓ At these times, \( a(t) > 0 \).
**(b)** Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time \( t = 0 \) to \( t = 7 \).
\[
\int_0^7 |\sin(\pi t / 3)| \, dt
\]
**(c)** Find the acceleration of the particle at time \( t = 2 \). (Round your answer to two decimal places.)
[Blank box for response]
**Explanation:**
The problem involves analyzing the motion of a particle along the x-axis with a specified velocity function, \( v(t) = \sin(\pi t / 3) \). Students are tasked with determining when the particle is moving to the right, justifying their reasoning based on the properties of velocity and acceleration, and formulating a mathematical expression for the distance traveled over a time interval. Additionally, they must calculate the acceleration at a specific time, exhibiting skills in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa89fcfbc-9661-4037-a401-8de41697cbc3%2Fdb9dbadd-5360-4297-a89a-fae60cf8eec0%2Fsgoe8toq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For educational purposes, here is the transcription and explanation of the image content:
**Problem Statement:**
Do not use technology.
For \( 0 \leq t \leq 7 \), a particle moves along the x-axis. The velocity of the particle is given by \( v(t) = \sin(\pi t / 3) \). The particle is at position \( x = -5 \) when \( t = 0 \).
**(a)** For \( 0 \leq t \leq 7 \), when is the particle moving to the right? (Enter your answer using interval notation.)
[Blank box for response]
**Justify your answer:**
- ○ At these times, \( x(t) < 0 \).
- ● At these times, \( v(t) > 0 \). (Selected)
- ○ At these times, \( v(t) < 0 \).
- ○ At these times, \( a(t) < 0 \).
- ○ At these times, \( x(t) > 0 \).
- ✓ At these times, \( a(t) > 0 \).
**(b)** Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time \( t = 0 \) to \( t = 7 \).
\[
\int_0^7 |\sin(\pi t / 3)| \, dt
\]
**(c)** Find the acceleration of the particle at time \( t = 2 \). (Round your answer to two decimal places.)
[Blank box for response]
**Explanation:**
The problem involves analyzing the motion of a particle along the x-axis with a specified velocity function, \( v(t) = \sin(\pi t / 3) \). Students are tasked with determining when the particle is moving to the right, justifying their reasoning based on the properties of velocity and acceleration, and formulating a mathematical expression for the distance traveled over a time interval. Additionally, they must calculate the acceleration at a specific time, exhibiting skills in calculus.
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