A laser light beamed at the Moon takes 2.56 s to return to earth after reflection at the Moon's surface. How much is the radius of the lunar orbit around the Earth ?

Question

UnlockIAS · Accepted Answer

Correct — option (4), 3.84 × 10⁸ m. The measurement described is laser ranging: a pulse is fired at the Moon, bounced off a reflector on its surface and detected again on Earth, and the time recorded is for the whole journey out and back. So the first step is to halve it. The round trip takes 2.56 seconds, therefore the one-way flight takes 1.28 seconds. The second step is to multiply by the speed of light, which is 3 × 10⁸ metres per second: distance = 3 × 10⁸ × 1.28 = 3.84 × 10⁸ metres. That is the Earth-Moon distance, which the stem calls the radius of the lunar orbit around the Earth, and it agrees with the accepted mean value of about 3.84 × 10⁵ kilometres — the same number, since one kilometre is 10³ metres. The option set demands one more piece of care than the arithmetic does, because it mixes units: the first and fourth rows are in metres and the second and third in kilometres, so nothing can be compared until everything is converted to a single unit. Convert the correct answer and it becomes 3.84 × 10⁵ km. Neither of the two kilometre rows carries that value: one reads 38.4 × 10⁵ km, which is 3.84 × 10⁶ km, and the other reads 76.8 × 10⁵ km, which is 7.68 × 10⁶ km. Each is ten times the metre-based value it otherwise resembles, so both are eliminated by conversion alone, and the whole of the work in this question lies in doing that conversion rather than in trusting the digits. The physical scale is worth carrying away as a check of its own. Light takes about 1.28 seconds to reach us from the Moon and about eight minutes and twenty seconds from the Sun, so any answer implying a Moon that light takes far longer to reach is describing something other than the Moon. Halving the round trip, multiplying by the speed of light and converting the units gives 3.84 × 10⁸ metres, which is option (4).

Why the other options are wrong — (a) This is the distance light travels in the whole 2.56 seconds — 3 × 10⁸ × 2.56 = 7.68 × 10⁸ metres — which is the round trip out to the Moon and back, not the distance to the Moon. It is the single commonest error in every echo-based measurement, whether the signal is laser light, radar or sonar, and it is worth building a permanent habit against: whenever a stem says a signal RETURNS, or is REFLECTED, or is an ECHO, halve the time before doing anything else. Write the halving down as the first line of the working rather than holding it in the head, because it is the step most easily lost when the arithmetic that follows is easy. The option is also the one a candidate is most likely to reach after a correct calculation performed too quickly, since every other element of the working is right and only the factor of two is missing — which is exactly why it is printed.  (b) This row makes two demands at once and fails both. Written out, 76.8 × 10⁵ km is 7.68 × 10⁶ kilometres. The round-trip distance in kilometres is 7.68 × 10⁵ km, so this value is not only the un-halved round trip rather than the distance to the Moon, it is also ten times that un-halved figure. Both faults have to be caught, and the way to catch them is to refuse to compare a row until it has been converted into the unit you have been working in. Bring it to metres and it becomes 7.68 × 10⁹ m, against a correct answer of 3.84 × 10⁸ m — different by a factor of twenty, which no reading of the stem can produce. The scale check settles it too: a Moon at seven and a half million kilometres would be some twenty times farther than the Moon actually is, and light would take about twenty-five seconds to reach it rather than the 1.28 seconds the stem itself implies.  (c) This is the most dangerous row in the set, because its digits are the right digits. Written out, 38.4 × 10⁵ km is 3.84 × 10⁶ kilometres, and the correct distance in kilometres is 3.84 × 10⁵ km — the same significant figures with the power of ten one step too high, which is to say ten times too far. A candidate who has done the halving and the multiplication correctly, arrived at 3.84 × 10⁸ metres, and then scanned the options for the familiar 3.84 will find it here and stop. The defence is mechanical: convert the row into your own working unit before accepting it, never the other way round. In metres this option reads 3.84 × 10⁹ m against a correct 3.84 × 10⁸ m, and the mismatch is then impossible to miss. When an option set mixes metres and kilometres, as this one does, it is testing conversion at least as much as it is testing the physics, and the correct digits appearing in the wrong row are the mechanism by which it does so.

Concept: Distances that cannot be walked out with a tape are measured by timing a signal of known speed, and the method is the same whether the signal is sound, radio or light. A pulse is sent, it reflects from the target, and the round-trip time is halved and multiplied by the speed of propagation. Sonar does this with sound in water to find the depth of the sea bed; radar does it with radio waves to find aircraft and to map the surfaces of Venus and the inner planets; laser ranging does it with light, and the Earth-Moon distance has been measured this way since retroreflector arrays were placed on the lunar surface — by the Apollo astronauts and by the Soviet Lunokhod rovers, which carried French-built reflectors. A retroreflector returns light back along the direction it came from, which is what makes the returning pulse detectable at all across such a distance. The speed of light in vacuum, about 3 × 10⁸ metres per second, is the constant that turns those times into distances, and it also supplies astronomy's larger units: light travels about 1.28 seconds from the Moon, about 8 minutes 20 seconds from the Sun, and one light year is the distance it covers in a year, roughly 9.46 × 10¹⁵ metres. The astronomical unit, the mean Earth-Sun distance, is about 1.496 × 10¹¹ metres. Lunar laser ranging is precise enough to have shown that the Moon is receding from the Earth by a few centimetres a year, a consequence of the tidal interaction between the two bodies.

State PCS 2021 Q56 — Science & Technology PYQ with Answer

State PCS 2021 Q56 — Science & Technology PYQ with Answer