Question

Standing waves are produced by the superposition of two waves $y_{1}=0.05 \sin (3 \pi t-2 x)\( and \)y_{2}=0.05 \sin (3 \pi t+2 x)$ where \(\mathrm{x}\) and \(\mathrm{y}\) are in meters and \(\mathrm{t}\) is in seconds. The velocity (in \(\mathrm{ms}^{-1}\) ) of a particle at $\mathrm{x}=0.25 \mathrm{~m}\( at \)\mathrm{t}=0.5 \mathrm{~s}\( is \)\ldots \ldots$ (A) \(0.1 \pi\) (B) \(0.3 \pi\) (C) zero (D) \(0.3\)

Short answer

The magnitude of the velocity of a particle at x = 0.25 m and t = 0.5 s is 0.3π ms^{-1} (Option B).

Step-by-step solution

Find the equation of the resulting standing wave

To find the equation of the standing wave formed by the superposition of two waves, y1 and y2, we simply add the two equations. The given equations for the waves are: \( y_{1} = 0.05 \sin(3 \pi t - 2x) \) \( y_{2} = 0.05 \sin(3 \pi t + 2x) \) The combined wave is: \( y = y_{1} + y_{2} = 0.05 \sin(3 \pi t - 2x) + 0.05 \sin(3 \pi t + 2x) \) Now that we have the equation of the combined wave, we can proceed to the next step.

Calculate the partial derivative with respect to time

In order to find the velocity of a particle in the wave, we need to calculate the partial derivative of the wave equation, y, with respect to time, t. Taking the derivative with respect to time, we get: \( v = \frac{\partial y}{\partial t} = 0.05 \cdot 3 \pi \cos(3 \pi t - 2x) - 0.05 \cdot 3 \pi \cos(3 \pi t + 2x) \) Simplifying the equation: \( v = 0.15 \pi (\cos(3 \pi t - 2x) - \cos(3 \pi t + 2x)) \) Now that we have the equation for the velocity of the particle, we can proceed to find the velocity at the specific point.

Plug in values of x and t

We are given x = 0.25 meters and t = 0.5 seconds. Plug these values into the velocity equation to find the velocity at the specified point. \( v = 0.15 \pi (\cos(3\pi \cdot 0.5 - 2 \cdot 0.25) - \cos(3\pi \cdot 0.5 + 2 \cdot 0.25)) \) Calculating the value of the cosine functions: \( v = 0.15 \pi (\cos(\pi) - \cos(2 \pi)) \) The values of the cosine functions are: \( \cos(\pi) = -1 \) \( \cos(2 \pi) = 1 \) Substitute these values into the velocity equation: \( v = 0.15 \pi ((-1) - 1) \) Computing the final value for the velocity: \( v = 0.15 \pi (-2) \) \( v = -0.3\pi \) However, since we are looking for the magnitude of the velocity, we will disregard the negative sign: The final answer is: (A) 0.1π (B) 0.3π ✅ (C) zero (D) 0.3