A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km/h. Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km/h. The average speed of the man over the interval of time 0 to 40 min. is equal to

The correct answer is Option 4: 45/8 km/h.

Let's break down the motion of the man step-by-step to find his average speed over the time interval from 0 to 40 minutes.

Step 1: Analyze the journey from home to the market
The distance from home to the market is:
$d=2.5km$
The speed of the man while walking to the market is:
$v_1=5km/h$
The time taken to reach the market ($t_1$) is given by the formula:
$t_1=\frac{\mathrm{Distance}}{\mathrm{Speed}}=\frac{2.5}{5}=0.5hours$
Converting this time into minutes:
$t_1=0.5\times 60min=30min$

Step 2: Analyze the return journey from the market back towards home
The total time interval we are interested in is 40 minutes (from $t=0$ to $t=40min$).
Since the man spent the first 30 minutes walking to the market, the remaining time for the return journey is:
$t_2=40min-30min=10min$
Converting this remaining time into hours:
$t_2=\frac{10}{60}hours=\frac{1}{6}hours$
During this return journey, the man walks back with a speed of:
$v_2=7.5km/h=\frac{15}{2}km/h$
The distance covered by the man in these 10 minutes ($d_2$) is:
$d_2=v_2\times t_2=7.5\times \frac{1}{6}=\frac{7.5}{6}=1.25km$

Step 3: Calculate the total distance covered and total time
The total distance covered by the man in the 40-minute interval is:
$d_{\mathrm{total}}=d_1+d_2=2.5km+1.25km=3.75km$
Expressing 3.75 as a fraction:
$d_{\mathrm{total}}=\frac{15}{4}km$
The total time interval is:
$t_{\mathrm{total}}=40min=\frac{40}{60}hours=\frac{2}{3}hours$

Step 4: Calculate the average speed
Average speed is defined as the ratio of the total distance covered to the total time taken:
$v_{\mathrm{avg}}=\frac{d_{\mathrm{total}}}{t_{\mathrm{total}}}$
Substituting the values we obtained:
$v_{\mathrm{avg}}=\frac{\frac{15}{4}}{\frac{2}{3}}=\frac{15}{4}\times \frac{3}{2}=\frac{45}{8}km/h$

Thus, the average speed of the man over the interval of 0 to 40 minutes is 45/8 km/h.