Q 11-14-144JEE MainJEE Main 2025 (7 Apr, Shift 2)Medium
The equation of a wave travelling on a string is $y = \sin(20\pi x + 10\pi t)$, where $x$ and $t$ are distance and time in SI units. The minimum distance between two points having the same oscillating speed (at all times) is:
Answer: (A) $5.0\ \text{cm}$
The particle velocity is $\dfrac{\partial y}{\partial t} = 10\pi\cos(20\pi x + 10\pi t)$. Two points always have the same speed (magnitude) when their phases differ by $\pi$, i.e. when they are $\lambda/2$ apart.
$k = 20\pi\ \text{m}^{-1}$, so $\lambda = \dfrac{2\pi}{k} = 0.1\ \text{m} = 10\ \text{cm}$, and the minimum distance is $\lambda/2 = 5\ \text{cm}$.
Solution by Sreeraj P, M.Sc Physics