Waves

JEE Main level, single correct, medium

23 of 142

Question 23 of 142

The equation of a plane progressive wave is given by y=5cos⁡π(200t−x150)y = 5\cos\pi\left(200t - \dfrac{x}{150}\right) where xx and yy are in cm and tt is in second. The velocity of the wave is _______ m/s.

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Answer: (D) 300300

The given equation of the wave is y=5cos⁡π(200t−x150)y = 5\cos\pi\left(200t - \dfrac{x}{150}\right)

Rewriting the equation, we get y=5cos⁡(200πt−πx150)y = 5\cos\left(200\pi t - \dfrac{\pi x}{150}\right)

Comparing this with the standard wave equation y=Acos⁡(ωt−kx)y = A\cos(\omega t - kx), we have:

ω=200π\omega = 200\pi rad/s

k=π150k = \dfrac{\pi}{150} rad/cm

The velocity of the wave is given by v=ωkv = \dfrac{\omega}{k}

v=200ππ150=30000v = \dfrac{200\pi}{\dfrac{\pi}{150}} = 30000 cm/s

Converting to m/s, we get v=30000100=300v = \dfrac{30000}{100} = 300 m/s

Answer: 300300

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