A traveling wave is represented by a wave function y(x, t)= (5.0 cm) sin (x-(0.20 m/s)t). When 3.0 m' t= 0 s, the first crest of this wave function on the positive side of the x coordinate origin is located at a. 0.25 m. b. 0.75 m. c. 1.50 m. d. 3.0 m. 3.0 m.
A traveling wave is represented by a wave function y(x, t)= (5.0 cm) sin (x-(0.20 m/s)t). When 3.0 m' t= 0 s, the first crest of this wave function on the positive side of the x coordinate origin is located at a. 0.25 m. b. 0.75 m. c. 1.50 m. d. 3.0 m. 3.0 m.
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Chapter1: Units, Trigonometry. And Vectors
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![**Traveling Wave Function and Crest Location**
A traveling wave is represented by a wave function given by:
\[ y(x, t) = (5.0 \, \text{cm}) \sin\left( \frac{2\pi}{3.0 \, \text{m}} \left( x - (0.20 \, \text{m/s}) t \right) \right). \]
**Problem Statement:**
When \( t = 0 \, \text{s} \), find the location of the first crest of this wave function on the positive side of the x-coordinate origin. The possible answer choices are:
a. 0.25 m
b. 0.75 m
c. 1.50 m
d. 3.0 m
e. \(\frac{3.0}{2\pi} \, \text{m} \)
**Explanation:**
At \( t = 0 \, \text{s} \):
\[ y(x, 0) = (5.0 \, \text{cm}) \sin\left( \frac{2\pi}{3.0 \, \text{m}} x \right). \]
To find the position of the first crest, we need to find the value of \( x \) that makes the argument of the sine function equal to \( \pi/2 \), because the sine of \( \pi/2 \) is 1 (which gives us the crest).
\[ \frac{2\pi}{3.0 \, \text{m}} x = \frac{\pi}{2} \]
Solving for \( x \):
\[ x = \frac{\pi}{2} \cdot \frac{3.0 \, \text{m}}{2\pi} = 0.75 \, \text{m} \]
**Answer:**
b. 0.75 m](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00822f5e-0c6f-4029-81e1-6a6ab50c835f%2F9dabebd8-f2dd-418c-9bfc-5487e78f163f%2F565w81b.jpeg&w=3840&q=75)
Transcribed Image Text:**Traveling Wave Function and Crest Location**
A traveling wave is represented by a wave function given by:
\[ y(x, t) = (5.0 \, \text{cm}) \sin\left( \frac{2\pi}{3.0 \, \text{m}} \left( x - (0.20 \, \text{m/s}) t \right) \right). \]
**Problem Statement:**
When \( t = 0 \, \text{s} \), find the location of the first crest of this wave function on the positive side of the x-coordinate origin. The possible answer choices are:
a. 0.25 m
b. 0.75 m
c. 1.50 m
d. 3.0 m
e. \(\frac{3.0}{2\pi} \, \text{m} \)
**Explanation:**
At \( t = 0 \, \text{s} \):
\[ y(x, 0) = (5.0 \, \text{cm}) \sin\left( \frac{2\pi}{3.0 \, \text{m}} x \right). \]
To find the position of the first crest, we need to find the value of \( x \) that makes the argument of the sine function equal to \( \pi/2 \), because the sine of \( \pi/2 \) is 1 (which gives us the crest).
\[ \frac{2\pi}{3.0 \, \text{m}} x = \frac{\pi}{2} \]
Solving for \( x \):
\[ x = \frac{\pi}{2} \cdot \frac{3.0 \, \text{m}}{2\pi} = 0.75 \, \text{m} \]
**Answer:**
b. 0.75 m
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