Question

A man walks on a straight road from his home to a market 2.5 \(\text {km}\) away with a speed of 5 \(\text {km}\) \(\text h^{-1}\). Finding the market closed, he instantly turns and walks back home with a speed of 7.5 \(\text {km}\) \(\text h^{-1}\) What is the 

1. magnitude of average velocity, and 
2. average speed of the man over the interval of time (i) 0 to 30 min, (ii) 0 to 50 min, (iii) 0 to 40 min ? [Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero !].

Answer 1

a. Time taken by the man to reach the market from home, \(\text t_1\) = \(\frac{2.5}{5}\) = \(\frac{1}{2}\) h = 30 min
Time taken by the man to reach home from the market, \(\text t_2\) = \(\frac{2.5}{7.5}\)= \(\frac{1}{3}\) h = 20 min
Total time taken in the whole journey = 30 + 20 = 50 min
Average velocity = \(\frac{\text{Displacement }}{\text{ Time}}\) = \(\frac{2.5}{\frac{1}{2}}\) = 5 km / h ....(a(i))

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Average Speed = \(\frac{\text{Distance }}{\text{ Time }}\)= \(\frac{2.5}{\frac{1}{2}}\) = 5 \(\text {km}/\text h\) ....(b(i))
Time = 50 min =\(\frac{5}{6}\) h
Net displacement = 0 
Total distance = 2.5 + 2.5 = 5\(\text {km}\)
Average velocity = \(\frac{\text{Displacement }}{\text{ Time }}\)= 0 ....(a(ii))
Average speed =\(\frac{\text{ Distance }}{\text{ Time}}\) = \(\frac{5}{\bigg(\frac{5}{6}\bigg)}\) = 6 \(\text {km}/\text h\).....(b(ii))
Speed of the man = 7.5 \(\text {km}\)
Distance travelled in first 30 min = 2.5 \(\text {km}\)
Distance travelled by the man (from market to home) in the next 10 min = 7.5 × 10 / 60 = 1.25 \(\text {km}\)
Net displacement = 2.5 – 1.25 = 1.25 \(\text {km}\)
Total distance travelled = 2.5 + 1.25 = 3.75 \(\text {km}\)
Average velocity = \(\frac{1.25}{\bigg(\frac{40}{60}\bigg)}\)= \(\frac{1.25\times 3}{2}\) = 1.875 \(\text {km}/\text h\)....(a(iii))
Average speed = \(\frac{3.75}{\bigg(\frac{40}{60}\bigg)}\) = 5.625 \(\text {km}/\text h\) ....(b(iii))