The average speed for a journey is the distance covered divided by the time taken.

(a) A journey is completed by travelling for the first half of the time at speed v₁ and the second half at speed v₂. Find the average speed v_a for the journey in terms of v₁ and v₂.

(b) A journey is completed by travelling at speed v₁ for half the distance and at speed v₂ for the second half. Find the average speed v_b for the journey in terms of v₁ and v₂. Deduce that a journey completed by travelling at two different speeds for equal distances will take longer than the same journey completed at the same two speeds for equal times.