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Waves question

2025 · 23 Jan · Shift 2 · Q51
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Waves question

2025 · 23 Jan · Shift 2 · Q51

JEE MainPhysicsWavesMCQ+4 / −1
The equation of a transverse wave travelling along a string is y(x,t)=4.0sin⁡[20×10−3x+600t]mmy(x, t)=4.0 \sin \left[20 \times 10^{-3} x+600 t\right] \mathrm{mm}y(x,t)=4.0sin[20×10−3x+600t]mm, where xxx is in mm and ttt is in second. The velocity of the wave is :
  1. A
    −60 m/s-60 \mathrm{~m} / \mathrm{s}−60 m/s
  2. B
    +60 m/s+60 \mathrm{~m} / \mathrm{s}+60 m/s
  3. C
    +30 m/s+30 \mathrm{~m} / \mathrm{s}+30 m/s
  4. D
    −30 m/s-30 \mathrm{~m} / \mathrm{s}−30 m/s
View written solutionFree

Correct answer: D

  1. Write the wave in standard form

A travelling wave is generally written as:

y(x,t)=Asin⁡(kx−ωt)orAsin⁡(kx+ωt)y(x,t)=A\sin(kx-\omega t) \quad \text{or} \quad A\sin(kx+\omega t)y(x,t)=Asin(kx−ωt)orAsin(kx+ωt)
  • Asin⁡(kx−ωt)A\sin(kx-\omega t)Asin(kx−ωt) represents motion in the +x+x+x direction.
  • Asin⁡(kx+ωt)A\sin(kx+\omega t)Asin(kx+ωt) represents motion in the −x-x−x direction.

Given:

y(x,t)=4.0sin⁡[20×10−3x+600t] mmy(x,t)=4.0\sin\left[20\times 10^{-3}x+600t\right] \text{ mm}y(x,t)=4.0sin[20×10−3x+600t] mm

So,

k=20×10−3 mm−1,ω=600 s−1k=20\times 10^{-3}\ \text{mm}^{-1}, \qquad \omega=600\ \text{s}^{-1}k=20×10−3 mm−1,ω=600 s−1
  1. Find the magnitude of wave speed

Wave speed is:

v=ωkv=\frac{\omega}{k}v=kω​

Substitute the values:

v=60020×10−3 mm/sv=\frac{600}{20\times 10^{-3}}\ \text{mm/s}v=20×10−3600​ mm/s v=6000.02=30000 mm/sv=\frac{600}{0.02}=30000\ \text{mm/s}v=0.02600​=30000 mm/s

Convert to m/s:

30000 mm/s=30 m/s30000\ \text{mm/s}=30\ \text{m/s}30000 mm/s=30 m/s
  1. Determine the direction

Since the wave is of the form

sin⁡(kx+ωt)\sin(kx+\omega t)sin(kx+ωt)

it travels in the negative xxx-direction.

Hence,

v=−30 m/sv=-30\ \text{m/s}v=−30 m/s
  1. Check options
  • A: −60 m/s-60\,\text{m/s}−60m/s ❌
  • B: +60 m/s+60\,\text{m/s}+60m/s ❌
  • C: +30 m/s+30\,\text{m/s}+30m/s ❌
  • D: −30 m/s-30\,\text{m/s}−30m/s ✅

Therefore, the correct answer is:

−30 m/s\boxed{-30\ \text{m/s}}−30 m/s​
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