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NEET PHYSICSMedium

The equation of a simple harmonic wave is given by y=3sin⁡π2(50t−x)y=3\sin\frac{\pi}{2}(50t-x) where xx and yy are in meters and tt is in seconds. The ratio of maximum particle velocity to the wave velocity is:

A

2π2\pi

B

32π\frac{3}{2}\pi

C

3π3\pi

D

23π\frac{2}{3}\pi

Step-by-Step Solution

  1. Identify the Given Wave Equation: The equation is y=3sin⁡π2(50t−x)y = 3\sin\frac{\pi}{2}(50t - x), which can be rewritten as y=3sin⁡(25πt−π2x)y = 3\sin(25\pi t - \frac{\pi}{2}x).
  2. Compare with the Standard Wave Equation: The standard progressive wave equation is y=Asin⁡(ωt−kx)y = A\sin(\omega t - kx) . By comparing, we get:
  • Amplitude, A=3 mA = 3 \text{ m}
  • Angular frequency, ω=25π rad/s\omega = 25\pi \text{ rad/s}
  • Wave number, k=π2 m−1k = \frac{\pi}{2} \text{ m}^{-1}
  1. Calculate Maximum Particle Velocity (vmax⁡v_{\max}): The maximum particle velocity is given by vmax⁡=Aω=3×25π=75π m/sv_{\max} = A\omega = 3 \times 25\pi = 75\pi \text{ m/s}.
  2. Calculate Wave Velocity (vv): The wave velocity is given by v=ωk=25ππ/2=50 m/sv = \frac{\omega}{k} = \frac{25\pi}{\pi/2} = 50 \text{ m/s} .
  3. Find the Ratio: The ratio of maximum particle velocity to the wave velocity is: vmax⁡v=75π50=32π\frac{v_{\max}}{v} = \frac{75\pi}{50} = \frac{3}{2}\pi (Alternatively, this ratio can be directly found as Ak=3×π2=32πAk = 3 \times \frac{\pi}{2} = \frac{3}{2}\pi).
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