Chapter 10, Problem 1IP

A runner completes 6½ laps around a 400 m track during a 12 min (720 s) run test. Calculate the following quantities:

a. The distance the runner covered

b. The runner’s displacement at the end of 12 min

c. The runner’s average speed

d. The runner’s average velocity

e. The runner’s average pace (Answers: a. 2.6 km; b. 160 m; c. 3.6 m/s; d. 0.22 m/s; e. 4.6 min/km)

Summary Introduction

To determine: The distance runner covered.

Answer to Problem 1IP

The distance covered by the runner is $2.6\text{km}$.

Explanation of Solution

Calculation:

Express the distance covered by the runner.

$\text{Distance}\text{covered}\text{by}\text{the}\text{runner}=\text{Number}\text{of}\text{laps}\times \text{distance}\text{in}\text{single}\text{lap}$

Substitute $6.5$ for number of laps and 400 m for distance in single lap to find the distance covered by the runner.

$\begin{array}{c}\text{Distance}\text{covered}\text{by}\text{the}\text{runner}=\text{6}\text{.5}\times \text{400}\text{m}\\ =\text{2600}\text{m}\left(\frac{1\text{km}}{1,000\text{m}}\right)\\ =2.6\text{km}\end{array}$

Therefore, distance covered by the runner is $2.6\text{km}$.

Conclusion

Therefore, distance covered by the runner is $2.6\text{km}$.

Summary Introduction

To determine: The runner’s displacement at the end of 12 min.

Answer to Problem 1IP

Displacement by the runner at the end of 12 min is $141.4\text{m}$.

Explanation of Solution

Calculation:

At 12 min runner completes 6.5 laps and the runner will be at the position of 0.5 lap.

Consider each side of the track is 100 m.

Use Pythagoras theorem to express the runner’s displacement at the end of 12 min.

$\begin{array}{l}\text{Runner’s}\text{displacement}\\ \text{at}\text{the}\text{end}\text{of}\text{12}\text{min}\end{array}\}\text{=}\sqrt{{\left(\text{Length}\text{of}\text{track}\right)}^2+{\left(\text{Breadth}\text{of}\text{track}\right)}^2}$

Substitute $100\text{m}$ for length and breadth to find the displacement by the runner at the end of 12 min.

$\begin{array}{c}\begin{array}{l}\text{Runner’s}\text{displacement}\\ \text{at}\text{the}\text{end}\text{of}\text{12}\text{min}\end{array}\}\text{=}\sqrt{{\left(\text{100}\right)}^2+{\left(\text{100}\right)}^2}\\ =141.4\text{m}\end{array}$

Therefore, displacement by the runner at the end of 12 min is $141.4\text{m}$.

Conclusion

Therefore, displacement by the runner at the end of 12 min is $141.4\text{m}$.

Summary Introduction

To determine: The runner’s average speed.

Answer to Problem 1IP

The average speed of the runner is $3.6\text{m}/\text{s}$.

Explanation of Solution

Calculation:

Express the average speed of the runner.

$\text{Average}\text{speed}=\frac{\text{Total}\text{distance}}{\text{total}\text{time}}$

Substitute 2600 m for total distance and 12 min to find the average speed.

$\begin{array}{c}\text{Average}\text{speed}=\frac{\text{2,600}\text{m}}{\text{12}\text{min}}\\ =\frac{\text{2,600}\text{m}}{\text{12}\text{min}\left(\frac{60\sec }{1\min }\right)}\\ =3.611\text{m}/\text{s}\\ \approx 3.6\text{m}/\text{s}\end{array}$

Therefore, average speed of the runner is $3.6\text{m}/\text{s}$.

Conclusion

Therefore, average speed of the runner is $3.6\text{m}/\text{s}$.

Summary Introduction

To determine: The runner’s average velocity.

Answer to Problem 1IP

The average velocity of the runner is $0.2\text{m}/\text{s}$.

Explanation of Solution

Calculation:

Express the average velocity of the runner.

$\text{Average}\text{velocity}=\frac{\text{Total}\text{displacement}}{\text{total}\text{time}}$

Substitute 141.4 m for total displacement and 12 min to find the average velocity.

$\begin{array}{c}\text{Average}\text{velocity}=\frac{\text{141}\text{.4}\text{m}}{\text{12}\text{min}}\\ =\frac{\text{141}\text{.4}\text{m}}{\text{12}\text{min}\left(\frac{60\sec }{1\min }\right)}\\ =0.19\text{m}/\text{s}\\ \approx 0.2\text{m}/\text{s}\end{array}$

Therefore, average velocity of the runner is $0.2\text{m}/\text{s}$.

Conclusion

Therefore, average velocity of the runner is $0.2\text{m}/\text{s}$.

Summary Introduction

To determine: The runner’s average pace.

Answer to Problem 1IP

The average pace of the runner is $4.6\text{min}/\text{km}$.

Explanation of Solution

Calculation:

Express the average pace of the runner.

$\text{Average}\text{pace}=\frac{\text{1}}{\text{Average}\text{speed}}$

Substitute $3.6\text{m}/\text{s}$ for average speed to find the average pace.

$\begin{array}{c}\text{Average}\text{pace}=\frac{\text{1}}{3.6\text{m}/\text{s}\left(\frac{\text{1}\text{km}}{\text{1,000}\text{m}}\right)\left(\frac{60\sec }{1\min }\right)}\\ =\frac{\text{1}}{0.2166\text{km}/\text{min}}\\ =4.61\text{min}/\text{km}\\ \approx 4.6\text{min}/\text{km}\end{array}$

Therefore, average pace of the runner is $4.6\text{min}/\text{km}$.

Conclusion

Therefore, average pace of the runner is $4.6\text{min}/\text{km}$.