3. Two children hold opposite ends of a rope and excite a sinusoidal travelling wave
in the vertical plane. A bystander notices that the time it takes for the rope to go from
its maximum height to its minimum height at any point along the rope is 0.5 s, and
that the distance between two peaks at any instant is 1.5 m. What is the speed of the
wave?
a. 1.5 m/s
b. 3 m/s
c. 1.5
m/s
d. 3 m/s
Sinusoidal Travelling Wave Speed on Rope
Two children generate a sinusoidal travelling wave on a rope, where the time from maximum to minimum height at any point is 0.5 s, and peak separation is 1.5 m. The wave speed is 1.5 m/s.
Wave Parameters Calculation
For a sinusoidal wave y = A sin(ωt – kx), maximum height occurs at ωt = π/2 and minimum at ωt = 3π/2, giving phase change Δ(ωt) = π.
With Δt = 0.5 s, ω = π/0.5 = 2π rad/s, so period T = 2π/ω = 1 s and frequency f = 1 Hz.
Wavelength λ equals distance between peaks, 1.5 m. Wave speed v = fλ = 1 × 1.5 = 1.5 m/s.
Options Analysis
| Option | Speed (m/s) | Explanation | Correct? |
|---|---|---|---|
| a | 1.5 | Matches v = fλ with T = 1 s, f = 1 Hz, λ = 1.5 m | Yes |
| b | 3 | Assumes T = 0.5 s (f = 2 Hz), but 0.5 s is half-period from max to min | No |
| c | 1.5 | Duplicate of option a; correct calculation | Yes |
| d | 3 | Uses full period as 0.5 s, ignoring quarter-to-three-quarter cycle timing | No |
Key Concept: Time from maximum to minimum displacement represents half the period (T/2 = 0.5 s), so T = 1 s.
