A car travels a total distance L. It travels half the distance with speed v₁ and the other half with speed v₂. The average speed of the car is:

Correct — B, 2v₁v₂/(v₁ + v₂). Average speed is defined as total distance divided by total time, so the whole problem is to add up the time. The first half of the journey, a distance L/2, takes (L/2)/v₁; the second half takes (L/2)/v₂. Adding them gives a total time of (L/2)(1/v₁ + 1/v₂), which is L(v₁ + v₂)/(2v₁v₂). Dividing the distance L by that time gives 2v₁v₂/(v₁ + v₂), the harmonic mean of the two speeds. Notice that L cancels — it has to, because how far you drive cannot change the average of two fixed speeds, and that single observation eliminates option (c) at once. Put numbers to it to see how far the answer sits from the obvious guess: at 60 km/h for the first half and 30 km/h for the second, the average is 2 × 60 × 30 ÷ 90 = 40 km/h, not 45. The slower half eats more of the clock, so it pulls the average down harder, and the harmonic mean is always the smaller of the two means unless the speeds are equal.

• (a)(v₁ + v₂)/2 — The arithmetic mean, and the answer to a different question. It would be right if the car spent half the TIME at each speed. Here it spends half the DISTANCE at each, so the slower stretch occupies more time and gets more weight.
• (c)(v₁ + v₂)L/(2v₁v₂) — This is the total time taken, not a speed at all — distance divided by speed. A quick check of units settles it, and so does the presence of L, since the average of two speeds cannot depend on the length of the trip.
• (d)0 — Zero is the answer for average VELOCITY on a round trip, where the displacement is nil. Average speed uses the path length actually covered, which is L here, and it is positive as long as the car moves.

Speed and velocity part company in exactly the way this question exploits. Average speed is the total path length divided by the total time; average velocity is the net displacement divided by the total time. Because the two share a denominator but not a numerator, average speed is never negative and is zero only if the body never moves, whereas average velocity vanishes whenever a body returns to where it started. Nothing in either definition allows you to average the individual speeds directly.

The pull towards option (a) is strong because averaging two numbers is what the word average suggests. The fix is to go back to the definition every time and build the total time first. There is a general shape here worth carrying: for equal distances the answer is the harmonic mean of the speeds, and for equal times the answer is the arithmetic mean. Two checks catch a wrong pick in seconds. One is dimensional — an expression containing L on its own cannot be a speed. The other is a limiting case: set v₁ = v₂ = v and a correct formula must collapse to v. Option (b) gives 2v²/2v = v, as it should, while option (c) gives L/v, which is a time.

• Average speed is total distance divided by total time, never the plain average of the individual speeds.
• For equal distances covered at two speeds, the average speed is the harmonic mean 2v₁v₂/(v₁ + v₂).
• For equal times spent at two speeds, the average speed is the arithmetic mean (v₁ + v₂)/2.
• The harmonic mean is always less than or equal to the arithmetic mean, with equality only when the two speeds are the same.
• Average velocity is displacement divided by time and is zero for any closed round trip, however fast the journey.

• Averaging the two speeds directly. That answers the equal-time version of the problem, not this one.
• Keeping L in the final expression; the answer cannot depend on how long the trip was.
• Reading the question as asking for average velocity and answering zero.

Either symbolically like this, or with numbers — up a hill at one speed and down at another, the return trip being the classic form.

• practice — not a real PYQ

  A cyclist covers the first half of a journey at 20 km/h and the second half at 30 km/h. What is the average speed for the whole journey?

  • (a)25 km/h
  • (b)24 km/h
  • (c)26 km/h
  • (d)22 km/h

  Answer(b) 24 km/h — equal distances call for the harmonic mean, 2 × 20 × 30 ÷ 50 = 24 km/h, which is less than the arithmetic mean of 25.

• practice — not a real PYQ

  A car drives for one hour at 40 km/h and the next hour at 60 km/h. Its average speed for the two hours is:

  • (a)48 km/h
  • (b)50 km/h
  • (c)45 km/h
  • (d)52 km/h

  Answer(b) 50 km/h — here the times are equal, not the distances, so the average is the arithmetic mean of the two speeds.