Chapter 7, Problem 81GP

To determine

The final speed of the aircraft.

Answer to Problem 81GP

The final speed of the aircraft is $-29.6\text{ km/s}$ .

Explanation of Solution

Given:

The orbital speed of Saturn is $v_{\text{s}}=-9.6\text{ km/s}$ .

The mass of Saturn is $m_{\text{s}}=5.69\times {10}^{26}\text{ kg}$ .

The mass of spacecraft is $m_{\text{sp}}=826\text{ kg}$ .

The initial speed of spacecraft is $v_{\text{sp}}=10.4\text{ km/s}$ .

Formula used:

The expression for relative velocity is,

  $u=v_{\text{sp}}-v_{\text{s}}$

Here, $v_{\text{sp}}$ is the velocity of spacecraft and $v_{\text{s}}$ is the velocity of Saturn.

Calculation:

The initial relative velocity is,

  $\begin{array}{l}{u}_{\text{sp}}=v_{\text{sp}}-v_{\text{s}}\\ u_{\text{sp}}=\left(10.4\text{km/s}\right)-\left(-9.6\text{km/s}\right)\\ u_{\text{sp}}=20\text{km/s}\end{array}$

The final velocity is,

  $\begin{array}{l}{u}_{\text{sp}}=v_{\text{sp,f}}-v_{\text{s}}\\ v_{\text{sp, f}}=\left(-20\text{km/s}\right)+\left(-9.6\text{km/s}\right)\\ v_{\text{sp, f}}=-29.6\text{km/s}\end{array}$

Conclusion:

Thus, the final speed of the aircraft is $v_{\text{sp,f}}=-29.6\text{ km/s}$ .