TTMUA Lab
TMUA 2018 · Paper 2 · Question 3 of 20

TMUA 2018 Paper 2 Question 3

Proof and counterexample — Counterexamples · average speed is not the mean of two speeds. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Proof and counterexampleCounterexamples · average speed is not the mean of two speeds8 options
Consider the following statement:

A car journey consists of two parts. In the first part, the average speed is u km/h. In the second part, the average speed is v km/h. Hence the average speed for the whole journey is 12(u+v) km/h.

Which of the following examples of car journeys provide(s) a counterexample to the statement?

I    In the first part of the journey, the car travels at a constant speed of 50 km/h for 100 km. In the second part of the journey, the car travels at a constant speed of 40 km/h for 100 km.
II   In the first part of the journey, the car travels at a constant speed of 50 km/h for one hour. In the second part of the journey, the car travels at a constant speed of 40 km/h for one hour.
III  In the first part of the journey, the car travels at a constant speed of 50 km/h for 80 km. In the second part of the journey, the car travels at a constant speed of 40 km/h for 100 km.

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · F
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Average speed is total distance over total time, so the claim only holds when the two parts take equal times. In II the times are equal by construction, so the average really is 45 km/h and it is not a counterexample. In I the distances are equal instead: the times are 2 h and 2.5 h, giving 200 ÷ 4.5 ≈ 44.4 km/h, which is not 45. In III the times are 1.6 h and 2.5 h, giving 180 ÷ 4.1 ≈ 43.9 km/h, again not 45. So I and III only.