A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 10t)

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Question: A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 10t)). What is the speed of the wave?

Options:

Correct Answer: 10 m/s

Solution:

The speed of the wave v is given by v = fλ. Here, f = 10 Hz and λ = 2 m (from k = 2π/λ). Thus, v = 10 * 2 = 20 m/s.

A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 10t)). What is the speed of the wave?

A wave on a string is described by the equation y(x, t) = 0.1 sin(2π(0.5x - 10t)). What is the speed of the wave?

Step 1: Identify the wave equation given: y(x, t) = 0.1 sin(2π(0.5x - 10t)).
Step 2: Recognize that the general form of a wave equation is y(x, t) = A sin(kx - ωt), where A is amplitude, k is the wave number, and ω is the angular frequency.
Step 3: From the equation, identify k and ω. Here, k = 2π * 0.5 and ω = 2π * 10.
Step 4: Calculate the wave number k: k = 2π * 0.5 = π.
Step 5: Calculate the angular frequency ω: ω = 2π * 10 = 20π.
Step 6: Find the frequency f using the formula f = ω / (2π): f = (20π) / (2π) = 10 Hz.
Step 7: Calculate the wavelength λ using the formula k = 2π / λ: λ = 2π / k = 2π / π = 2 m.
Step 8: Use the wave speed formula v = f * λ: v = 10 Hz * 2 m = 20 m/s.

Wave Equation Analysis – Understanding the wave equation and identifying parameters such as frequency and wavelength.
Wave Speed Calculation – Using the relationship between wave speed, frequency, and wavelength to calculate the speed of the wave.