Chapter 7.3, Problem 97PFA

a.

To determine

To calculate the time taken by the space craft.

Answer to Problem 97PFA

18 seconds

Explanation of Solution

Given:

Formula: $t=\frac{2\pi r^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ where t is the time, g is the gravity and ris the radius.

  $r=80$ , $g=9.8$

Calculation:

Given that the time is modeled by $t=\frac{2\pi r^{\frac{1}{2}}}{g^{\frac{1}{2}}}$

Substituting the values of r and g,

  $\begin{array}{l}\Rightarrow t=\frac{2\pi r^{\frac{1}{2}}}{g^{\frac{1}{2}}}\\ \Rightarrow t=\frac{2\pi {80}^{\frac{1}{2}}}{{9.8}^{\frac{1}{2}}}\\ \Rightarrow t=\frac{2\pi (8.94)}{3.13}\\ \Rightarrow t=\frac{17.88\pi }{3.13}\\ \Rightarrow t=\frac{56.14}{3.13}\\ \Rightarrow t=17.93\end{array}$

Conclusion:

Therefore the time taken is approximately 18 seconds.

b.

To determine

To calculate the gravity after 20 seconds.

Answer to Problem 97PFA

  $g=8.82m/s^2$

Explanation of Solution

Given:

Formula: $t=\frac{2\pi r^{\frac{1}{2}}}{g^{\frac{1}{2}}}$ where t is the time, g is the gravity and ris the radius.

  $r=90$ , $t=20$

Calculation:

Given that the time is modeled by $t=\frac{2\pi r^{\frac{1}{2}}}{g^{\frac{1}{2}}}$

Substituting the values of r and t,

  $\begin{array}{l}\Rightarrow t=\frac{2\pi r^{\frac{1}{2}}}{g^{\frac{1}{2}}}\\ \Rightarrow g^{\frac{1}{2}}=\frac{2\pi r^{\frac{1}{2}}}{t}\\ \Rightarrow g^{\frac{1}{2}}=\frac{2\pi {90}^{\frac{1}{2}}}{20}\\ \Rightarrow g^{\frac{1}{2}}=\frac{2\pi 9.48}{20}\\ \Rightarrow g^{\frac{1}{2}}=\frac{18.96\pi }{20}\\ \Rightarrow g^{\frac{1}{2}}=\frac{59.53}{20}\\ \Rightarrow g^{\frac{1}{2}}=2.97\\ \Rightarrow g=8.82m/s^2\end{array}$

Conclusion:

Therefore, the gravity at 90 meters after 20 seconds is $g=8.82m/s^2$