Chapter 2, Problem 70GP

To determine

The average speed maintained for the last lap

Answer to Problem 70GP

  $220\frac{\text{km}}{\text{h}}$

Explanation of Solution

Given:

Distance travelled in each lap $=d$

Distance travelled for ten laps $=d_{10}$

Time taken for ten laps $=t_{10}$

Average speed for ten laps $=v_{10}=200\frac{\text{km}}{\text{h}}$

Distance travelled for nine laps $=d_9$

Time taken for nine laps $=t_9$

Average speed for nine laps $=v_9=198\frac{\text{km}}{\text{h}}$

Average speed for last lap $=v_{last}$

Time taken for last lap $=t_{last}$

Formula Used:

Average speed is given as

  $\text{average speed =}\frac{\text{Total distance}}{\text{Total time}}$

Calculation:

Distance travelled in ten laps is given as

  $d_{10}=10d$

Average speed for ten laps is given as

  $\begin{array}{l}{v}_{10}=\frac{d_{10}}{t_{10}}\\ 200=\frac{10d}{t_{10}}\\ t_{10}=\frac{10d}{200}\left(1\right)\end{array}$

Distance travelled in nine laps is given as

  $d_9=9d$

Average speed for nine laps is given as

  $\begin{array}{l}{v}_9=\frac{d_9}{t_9}\\ 198=\frac{9d}{t_9}\\ t_9=\frac{9d}{198}\left(\text{2}\right)\end{array}$

Distance travelled in last lap is given as

  $d_{last}=d$

Time taken for last lap is given as

  $\begin{array}{l}{t}_{last}=t_{10}-t_9\\ \text{Using equation }\left(1\right)\text{ and }\left(2\right)\\ t_{last}=\frac{10d}{200}-\frac{9d}{198}\left(3\right)\end{array}$

Average speed for the last lap is given as

  $v_{last}=\frac{d_{last}}{t_{last}}$

Using equation (3),

  $\begin{array}{l}{v}_{last}=\frac{d}{\left(\frac{10d}{200}-\frac{9d}{198}\right)}\\ v_{last}=\frac{1}{\left(\frac{10}{200}-\frac{9}{198}\right)}\\ v_{last}=220\frac{\text{km}}{\text{h}}\end{array}$

Conclusion:

Therefore, the average speed for the last lap is $220\frac{\text{km}}{\text{h}}$