Question
On a 20-mile bike ride, you ride the first 10 miles at an average speed of 8 mi/h.
(a) What must your average speed over the next 10 miles be in order to have a total average speed of 4 mi/h?
(b) What must your average speed over the next 10 miles be in order to have a total average speed of 12 mi/h?
(c) Given your answer in part (b), is it possible to attain a total average speed of 16 mi/h for the entire 20-mile ride? Explain.
Solution
Let us solve Problem 2.54 step by step.
Given data:
• The total bike ride distance = 20 miles
• First 10 miles average speed = 8 mi/h
• We need to find the required average speed for the second 10 miles to achieve certain total average speeds.
Step 1: Definition of Average Speed
The average speed over the entire trip is given by:
Average speed = total distance/ total time
where Total time is:
Ttotal = t1 + t2
with:
• t1 = time taken for first 10 miles
• t2 = time taken for second 10 miles
The time for a given distance at speed is: t = d/v
Step 2: Time for First 10 Miles
t1 = 10/8 = 1.25 hours
Step 3: Find Required Speed for Given Average Speeds
(a) Required speed for total average speed of 4 mi/h
20/(t1 + t2) = 4
Substituting t1 = 1.25 hours
t2 = 5 – 1.25 = 3.75 hours
v2 = 10/t2 = 10/3.75 =2.667 mi/h
So, the required speed is 2.67 mi/h.
(b) Required speed for total average speed of 12 mi/h
20/(t1 + t2) = 12
ttotal = 20/12 = 1.667 hours
t2 = 1.67 – 1.25 = 0.42 hours
v2 = 10/0.42 = 23.81 mi/h
So, the required speed is 23.8 mi/h.
(c) Is it possible to attain an average speed of 16 mi/h for the total 20 miles?
20/(t1 + t2) = 16
Ttotal = 20/16 = 1.25 hours
t2 = 1.25 – 1.25 = 0
Since time cannot be zero, it is impossible to achieve an average speed of 16 mi/h for the total trip.
Final Answers
(a) Required speed = 2.67 mi/h
(b) Required speed = 23.8 mi/h
(c) Not possible to achieve 16 mi/h average speed.