average speed

Question

A car travels the first half of a distance at 40 km/h and the second half at 60 km/h. What is the average speed for the whole journey?

RankGuru · Accepted Answer

1. **Common Trap**: Average speed is NOT necessarily the arithmetic mean $(40+60)/2 = 50$.
2. **Definition**: Average speed $v_{avg} = \frac{	ext{Total distance}}{	ext{Total time}}$.
3. **Setup**: Let half the distance be $d$.
   - Time $t_1 = d / 40$.
   - Time $t_2 = d / 60$.
4. **Total Time**:
   - $t_{total} = \frac{d}{40} + \frac{d}{60} = \frac{3d + 2d}{120} = \frac{5d}{120} = \frac{d}{24}$.
5. **Final Velocity**:
   - $v_{avg} = \frac{2d}{d/24} = 48$ km/h.
6. **Shortcut**: Use harmonic mean formula: $v_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2 	imes 40 	imes 60}{100} = 48$ km/h.

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