average vs instantaneous speed

Question

A car moves the first 30 km at 60 km/h and the next 30 km at 90 km/h. What is the average speed for the whole trip?

RankGuru · Accepted Answer

1. **Definition of Average Speed ($v_{avg}$)**: $v_{avg} = \frac{	ext{Total Distance}}{	ext{Total Time}}$.
2. **Time for Segment 1**: $t_1 = \frac{d_1}{v_1} = \frac{30 	ext{ km}}{60 	ext{ km/h}} = 0.5$ h.
3. **Time for Segment 2**: $t_2 = \frac{d_2}{v_2} = \frac{30 	ext{ km}}{90 	ext{ km/h}} = \frac{1}{3} \approx 0.333$ h.
4. **Total Trip Analysis**:
   - Total Distance $= 30 + 30 = 60$ km.
   - Total Time $= 0.5 + 0.333 = 0.833$ h or $(\frac{1}{2} + \frac{1}{3} = \frac{5}{6})$ h.
5. **Calculate Speed**: $v_{avg} = \frac{60}{5/6} = 60 	imes \frac{6}{5} = 12 	imes 6 = 72$ km/h.
6. **Key Insight**: Average speed is the harmonic mean of speeds when distances are equal: $v_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2 	imes 60 	imes 90}{60+90} = \frac{10800}{150} = 72$ km/h.

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average vs instantaneous speed

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