The equation of a wave traveling in a horizontal direction is expressed as y = 15 sin 2π5 (60t - x). What is its wavelength?

Physics

ScanSolve AI Answer

Answer

0.2

Step-by-step solution

5 steps
  1. Identify the general form of a wave equation.

    The general equation for a sinusoidal wave traveling in the positive x-direction is given by:

    y=Asin⁡(ωt−kx)y = A \sin(\omega t - kx)

    where AA is the amplitude, ω\omega is the angular frequency, and kk is the wave number.

  2. Rewrite the given equation in the standard form.

    The given equation is y=15sin⁡(2π⋅5(60t−x))y = 15 \sin \left(2\pi \cdot 5 (60t - x)\right).
    First, distribute the 2π⋅52\pi \cdot 5 term inside the parenthesis:

    y=15sin⁡((2π⋅5⋅60)t−(2π⋅5)x)y = 15 \sin \left( (2\pi \cdot 5 \cdot 60)t - (2\pi \cdot 5)x \right) y=15sin⁡(600πt−10πx)y = 15 \sin \left( 600\pi t - 10\pi x \right)
  3. Compare with the standard form to find the wave number kk.

    By comparing y=15sin⁡(600πt−10πx)y = 15 \sin \left( 600\pi t - 10\pi x \right) with y=Asin⁡(ωt−kx)y = A \sin(\omega t - kx), we can identify:
    ω=600π\omega = 600\pi
    k=10πk = 10\pi

  4. Calculate the wavelength λ\lambda using the wave number kk.

    The relationship between the wave number kk and the wavelength λ\lambda is:

    k=2πλk = \frac{2\pi}{\lambda}

    Substitute the value of kk we found:

    10π=2πλ10\pi = \frac{2\pi}{\lambda}
  5. Solve for λ\lambda.

    Divide both sides by 2π2\pi:

    λ=2π10π\lambda = \frac{2\pi}{10\pi} λ=210\lambda = \frac{2}{10} λ=0.2\lambda = 0.2

    The wavelength of the wave is 0.2 units.

    The final answer is 0.2.

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