KCET · Physics · Waves and Sound
\(\mathrm{y}=3 \sin \pi\left(\frac{\mathrm{t}}{2}-\frac{\mathrm{x}}{4}\right)\) represents an equation of a progressive wave, where \(t\) is in second and \(x\) is in metre. The distance travelled by the wave in \(5 \mathrm{~s}\) is
- A \(8 \mathrm{~m}\)
- B \(10 \mathrm{~m}\)
- C \(5 \mathrm{~m}\)
- D \(32 \mathrm{~m}\)
Answer & Solution
Correct Answer
(B) \(10 \mathrm{~m}\)
Step-by-step Solution
Detailed explanation
The given equation of a progressive wave is \(y=3 \sin \pi\left(\frac{t}{2}-\frac{x}{4}\right)=3 \sin 2 \pi\left(\frac{t}{4}-\frac{x}{8}\right)\)
The standard equation of a progressive wave is
\(\mathrm{y}=\mathrm{y}_{0} \sin 2 \pi\left(\frac{\mathrm{t}}{\mathrm{T}}-\frac{\mathrm{x}}{\lambda}\right)\)
Comparing these two equations, we get
\(\mathrm{T}=4 \mathrm{~s}, \lambda=8 \mathrm{~m}\)
\(\therefore\) Velocity of wave,
\(\mathrm{v}=\frac{\lambda}{\mathrm{T}}=\frac{8}{4}=2 \mathrm{~ms}^{-1}\)
Distance travelled by wave in time \(t\) is
\(\mathrm{s}=\mathrm{vt}\)
or
\(\mathrm{s}=2 \times 5=10 \mathrm{~m}\)
The standard equation of a progressive wave is
\(\mathrm{y}=\mathrm{y}_{0} \sin 2 \pi\left(\frac{\mathrm{t}}{\mathrm{T}}-\frac{\mathrm{x}}{\lambda}\right)\)
Comparing these two equations, we get
\(\mathrm{T}=4 \mathrm{~s}, \lambda=8 \mathrm{~m}\)
\(\therefore\) Velocity of wave,
\(\mathrm{v}=\frac{\lambda}{\mathrm{T}}=\frac{8}{4}=2 \mathrm{~ms}^{-1}\)
Distance travelled by wave in time \(t\) is
\(\mathrm{s}=\mathrm{vt}\)
or
\(\mathrm{s}=2 \times 5=10 \mathrm{~m}\)
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