The equation of a wave travelling on a string is y = sin[20πx + 10πt], where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is :

Solution

Physics sir · 4 steps

Speed ignores direction, so the pattern of speeds repeats every half wavelength, not every full one.

1. Read off λ. Compare with y = A sin(kx + ω t): k = 20π m⁻¹. So λ = 2π/k = 2π/20π = 0.1 m = 10 cm.
2. Particle velocity. Differentiate y with respect to t to get the velocity of the particle at position x.

   v = (∂ y)/(∂ t) = 10π cos(20π x + 10π t)

3. Half wavelength apart. Points that always share the same speed need |cos θ| to match at every instant. Since cos(θ + π) = -cos θ, a phase gap of π is the smallest that works: 20π Δ x = π, so Δ x = 1/20 m = λ/2.
4. So. The two points move with equal speed in opposite directions at all times, and the minimum separation is λ/2 = 0.05 m = 5.0 cm, option (d).

The idea to keep. Same velocity (with direction) repeats every λ. Same speed (magnitude only) repeats every λ/2, because points in opposite phase move equally fast but opposite ways.

Answer from the NTA final answer key. Solution by PrepWiser.