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Question

The equation of a tranverse wave propagating through a medium is given by y=10sin(πt+0.5x). Choose the correct statement for the velocity of particle present at x=0.

A
At t=0, velocity of particle is 0 m/s
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B
At t=0.5 sec, velocity of particle is π m/s
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C
At t=3 sec, velocity of particle is 10π m/s
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D
At t=0.5 sec, velocity of particle is 0 m/s
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Solution

The correct option is D At t=0.5 sec, velocity of particle is 0 m/s
The displacement equation for the wave is,
y=10sin(πt+0.5x)
Differentiating above equation partially w.r.t time gives velocity of particle at a fixed position.
vP=yt
vP=π×10cos(πt+0.5x)
Substituting x=0 in above equation to get particle velocity at x=0,
vP=π×10cos(πt+0)
vP=10πcosπt

At t=0 sec, particle velocity is
vP=10π×cos0=10π m/s

At t=3 sec
vP=10πcos(3π)=10π m/s
similarly, at t=0.5 sec
vP=10πcos(π/2)=0 m/s

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