A pendulum has a frequency of 4 vibrations per second. An observer starts the pendulum and fires a gun simultaneously. He hears the echo from the cliff after 6 vibrations of the pendulum. If the velocity of sound in air is 340 m s-1, find the distance between the cliff and the observer.

Given: pendulum frequency = 4 vibrations per second, echo heard after 6 vibrations

speed of sound v = 340 m s⁻¹

Time for one vibration = 1 ÷ 4 = 0.25 s Time for 6 vibrations, t = 6 × 0.25 = 1.5 s

In that time the sound travels to the cliff AND back, so 2d = v × t 2d = 340 × 1.5 2d = 510 d = 510 ÷ 2 d = 255 m

Distance between the cliff and the observer = 255 m

Two steps, and the pendulum is only a clock. Turn the vibrations into seconds first: 4 vibrations per second means each one takes 0.25 s, so 6 of them take 1.5 s.

Then it is an ordinary echo problem, so the sound covers 2d. Forgetting the factor of 2 gives 510 m — the single most common error in this topic.

Multiplying 6 by 4 instead of dividing is the other slip; that would give 24 s, and an absurd distance of 4 km. Ask whether 6 vibrations should take more or less than a second: at 4 per second, six of them must take a little over one second.