CDS (II) 2021 General Knowledge · Question 19 of 120

Correct — A, 2·0 s. Work out the speed first and the time falls out. Speed of a wave is its wavelength multiplied by its frequency, so with a frequency of 4 kHz — that is 4000 hertz — and a wavelength of 30 cm, which is 0·30 m, the speed comes to 4000 × 0·30 = 1200 metres per second. The journey is 2·4 km, which is 2400 m. Time is distance divided by speed, so 2400 ÷ 1200 = 2 s. Two habits keep this from going wrong. Convert everything into metres and seconds before multiplying, since a stray centimetre or kilometre is the usual source of a factor of a hundred or a thousand. And notice that the speed obtained, 1200 m/s, is a plausible one — faster than sound in air at about 343 m/s but slower than in steel at nearly 6000 m/s — which is a quick sanity check that the arithmetic has not slipped a decimal place.

• (b)0·6 s — Comes out if the distance is divided by a speed that is too large by a factor of a little over three. There is no step in the working that produces such a speed.
• (c)1·0 s — Would need a speed of 2400 m/s, which is double the wave's actual speed. It usually appears when the wavelength is read as 60 cm or the distance halved by mistake.
• (d)8·0 s — Four times the right answer, and the sort of value that appears when 30 cm is left in centimetres or 4 kHz is treated as 1000 hertz.

Every wave obeys the same relation between its three descriptors: speed equals wavelength times frequency. Frequency is fixed by the source and does not change when the wave enters a new medium; speed depends chiefly on the nature and the temperature of the medium; wavelength adjusts so that the product still works out. Time for a wave to cover a stated distance is then simply that distance divided by the speed.

This is a two-step numerical dressed up as a one-line question, and almost every wrong answer on offer comes from a unit slip rather than a conceptual error. Write out kilo as a thousand, centi as a hundredth, and the sum is safe. The item is a close reworking of a standard textbook worked example, which used 2 kHz, 35 cm and 1·5 km; the NDA paper of 2022 set the same problem with 1 kHz, 50 cm and 1 km. Recognising the pattern is worth as much as the arithmetic. Note the printing convention in the paper: the decimal point is set as a raised dot, so 2·4 km means two point four kilometres and 2·0 s means two point zero seconds. It is a typographic style, not a range or a multiplication sign, and it is reproduced here exactly as the booklet printed it.

• Speed of a wave equals its wavelength multiplied by its frequency.
• 4 kHz is 4000 hertz and 30 cm is 0·30 m, giving a wave speed of 1200 m/s.
• 2·4 km is 2400 m, so the time taken is 2400 divided by 1200, which is 2 s.
• The speed of sound depends primarily on the nature and the temperature of the transmitting medium, and is about 343 m/s in air at 20 °C.
• Frequency is set by the source and stays the same when a wave moves into a new medium; the speed and the wavelength are what change.

• Leaving the wavelength in centimetres or the distance in kilometres before multiplying or dividing.
• Reading 4 kHz as 4 hertz, or as 400 hertz.
• Mistaking the raised dot in the printed numbers for anything other than a decimal point.

As a two-step numerical like this one, as a direct application of speed equals wavelength times frequency, or as a comparison of the speed of sound in different media.

• practice — not a real PYQ

  A sound wave of frequency 500 Hz has a wavelength of 0·66 m in air. Its speed in air is

  • (a)330 m/s
  • (b)660 m/s
  • (c)165 m/s
  • (d)1320 m/s

  Answer(a) 330 m/s — speed equals wavelength times frequency, that is 0·66 × 500.

• practice — not a real PYQ

  When a sound wave travels from air into water, which one of the following remains unchanged?

  • (a)Its speed
  • (b)Its wavelength
  • (c)Its frequency
  • (d)Its intensity

  Answer(c) Its frequency — frequency is fixed by the source, while speed and wavelength both change with the medium.