Chapter 2.4, Problem 16E

To determine

To calculate: The time taken by bullet to return.

Answer to Problem 16E

The time taken in case of moon is $t=320\sec$ and time taken in case of earth is $t=52\sec$ .

Explanation of Solution

Given information:

The position reached from moon is $s=832t-2.6t^2$ and position reached from earth is $832t-16t^2$

Calculation:

From moon:

Differentiate $s=832t-2.6t^2$ with respect to time $t$ .

  $\begin{array}{c}\frac{ds}{dt}=\frac{d}{dt}\left(832t-2.6t^2\right)\\ =\frac{d}{dt}\left(832t\right)-\frac{d}{dt}\left(2.6t^2\right)\\ =832-5.2t\end{array}$

To find the time equate $832-5.2t$ to zero.

  $\begin{array}{c}832-5.2t=0\\ 5.2t=832\\ t=\frac{832}{5.2}\\ t=160\sec \end{array}$

The above time is only the time required to travel half of the distance to get the total time required multiply the above obtained time by $2$ .

The time taken to get to moon is $160\times 2=320\sec$ .

From earth:

Differentiate $832t-16t^2$ with respect to time $t$ .

  $\begin{array}{c}\frac{ds}{dt}=\frac{d}{dt}\left(832t-16t^2\right)\\ =\frac{d}{dt}\left(832t\right)-\frac{d}{dt}\left(16t^2\right)\\ =832-32t\end{array}$

To find the time equate $832-32t$ to zero.

  $\begin{array}{c}832-32t=0\\ 32t=832\\ t=\frac{832}{32}\\ t=26\sec \end{array}$

The above time is only the time required to travel half of the distance to get the total time required multiply the above obtained time by $2$ .

The time taken to get to earth is $26\times 2=52\sec$ .

Conclusion: So, the time get back from moon is $t=320\sec$ and from earth is $t=52\sec$ .