A boy completes one round of a circular track of diameter 200 m in 30 s. What will be the displacement at the end of 3 minutes and 45 seconds?
- (a)50 m
- (b)100 m
- (c)200 m
- (d)236 m
Correct — C, 200 m. Displacement is the straight line from where the boy started to where he ends up, so the only thing that matters is where on the circle he stops. Three minutes and forty-five seconds is 225 seconds, and at 30 seconds a lap that is exactly 7·5 laps. The seven whole laps bring him back to the starting point and contribute nothing; the remaining half lap carries him to the point diametrically opposite, and the straight-line distance across a circle from one end of a diameter to the other is the diameter itself, 200 m. The distance he has actually run over the same time is a very different number — 7·5 laps of circumference π × 200 m comes to about 4712 m — and the question is built on keeping the two apart.
- (a)50 m — A quarter of the diameter, with no step in the working that produces it. A quarter lap would give a chord of about 141 m, not 50 m, and in any case the elapsed time works out to a clean half lap.
- (b)100 m — The radius. It is what comes out if the figure of 200 m in the stem is read as a radius rather than a diameter, or if the diameter is halved by reflex at the end. The stem says diameter, and the far end of a diameter is a full 200 m away.
- (d)236 m — Impossible for this track at any time whatever. The greatest separation between two points on a circle is its diameter, so no displacement on a 200 m-diameter track can exceed 200 m. A number larger than the diameter can be rejected before any arithmetic is done.
Distance is a scalar — the whole length of the path travelled, which only ever grows. Displacement is a vector — the straight line from the starting point to the finishing point, with a direction, and it can grow, shrink or return to zero. On a circular track the two part company at once: after every whole lap the displacement is zero while the distance keeps climbing. The displacement is largest after an odd number of half laps, when it equals the diameter, and it can never exceed the diameter, because no two points on a circle are further apart than that.
Two habits solve this. Convert the time into laps before anything else — 225 seconds divided by 30 seconds gives 7·5 — and then discard the whole-number part, since whole laps return the boy to where he began and leave the displacement untouched. That reduces the problem to 'where does half a lap put him', and the answer is the far end of a diameter. The upper-bound check is worth keeping too, because it kills one option outright. As a piece of physics the numbers are an exam abstraction rather than a description of anybody real: a lap of this track is about 628 m, and covering it in 30 seconds means an average speed near 21 m/s, which is faster than the quickest human being has ever run.
- Distance is the length of the path travelled; displacement is the straight line from start to finish and carries a direction.
- 3 minutes 45 seconds is 225 seconds, which at 30 seconds a lap is 7·5 laps.
- After a whole number of laps the displacement is zero; after a half lap it equals the diameter of the track.
- The displacement on a circular track can never exceed the diameter, however long the runner keeps going.
- The distance covered here is 7·5 × π × 200 m, roughly 4712 m — about twenty-three times the displacement.
- Computing the distance travelled and offering it as the displacement.
- Forgetting to convert minutes and seconds into a single number of seconds before dividing.
- Reading a stated diameter as a radius, which halves or doubles every answer that follows.
Almost always as a circular-track item with a time that works out to a whole-and-a-half number of laps, asking for displacement, average velocity, or both together.
CDS_GK_2021_II_Q62021An athlete completes one round of a circular track of diameter 100 m in 20 s. What will be the displacements after 1 minute and 10 s, respectively ?
- (a) 0 m, 50 m
- (b) 300 m, 100 m
- (c) 300 m, 50 m
- (d) 0 m, 100 m
Answer(d) 0 m, 100 m
The same question with the numbers changed, set six months earlier. One minute is three whole rounds and gives zero; ten seconds is half a round and gives the diameter. Exactly the same two-step method answers both.
- practice — not a real PYQ
A cyclist rides four complete rounds of a circular park of radius 70 m. Her average velocity for the ride is
- (a)equal to her average speed
- (b)zero
- (c)440 m per lap time
- (d)1760 m divided by the total time
Answer(b) zero — after four whole rounds she is back at the starting point, so the displacement is zero and average velocity, which is displacement divided by time, is zero as well.
- practice — not a real PYQ
An athlete runs exactly one and a half rounds of a circular track of radius 50 m. The magnitude of his displacement is
- (a)zero
- (b)50 m
- (c)100 m
- (d)471 m
Answer(c) 100 m — the whole round contributes nothing and the extra half round leaves him at the opposite end of a diameter, which is 2 × 50 m across.