Chapter 6.6, Problem 28E

(a).

To determine

The work needed to propel a satellite out of the earth’s gravitational field.

Answer to Problem 28E

The work needed to propel the satellite out of the earth’s gravitational is $6.26\times {10}^{10}J.$

Explanation of Solution

Given:

• Mass of the satellite is $1000kg$ .
• Mass of the earth is $5.98\times {10}^{24}kg$ .
• Radius of the earth is $6.37\times {10}^{6}m$ .
• The satellite’s final orbit is out of the earth’s gravitational field.
• The value of $G$ is $6.67\times {10}^{-11}N\cdot m^2/k{g}^2$ .

Concept Used: The work needed to launch a satellite of mass $m$ from the surface of the earth (radius $=R,mass=M$ ) and place it in an orbit of radius $r$ around the earth is $W_{R\to r}=GmM\left(\frac{1}{R}-\frac{1}{r}\right)$

Calculation:The value of the final radius $r$ is $\infty$ because the satellite is far away from the earth and out of its gravitational field.

So, put $r=\infty$ and thegiven values of $G,m,M,R$ in the expression for work to get

  $W_{R\to \infty }=6.67\times {10}^{-11}\times {10}^3\times 5.98\times {10}^{24}\left(\frac{1}{6.37\times {10}^6}-\frac{1}{\infty }\right)$

  $=6.67\times 5.98\times {10}^{\left(-11+3+24-6\right)}\left(\frac{1}{6.37}-0\right)$

  $=6.26\times {10}^{10}$

Therefore, the work needed to propel the satellite out of the earth’s gravitational field is $6.26\times 10^{10}J$

(b).

To determine

The escape velocity of a rocket on planet.

Answer to Problem 28E

Explanation of Solution

Given:

• Mass of the rocket is $m$ .
• Mass of the planet is $M$ .
• Radius of the earth is $R$ .
• The rocket’s final position is out of the earth’s gravitational field.
• The required work for propulsion is entirely provided by the initial kinetic energy $\frac{1}{2}m{v}_e^2$ where $v_e$ is the escape velocity.

Concept Used: The work needed to launch a satellite of mass $m$ from the surface of the earth (radius $=R,mass=M$ ) and place it in an orbit of radius $r$ around the earth is $W_{R\to r}=GmM\left(\frac{1}{R}-\frac{1}{r}\right)$

Calculation: The value of the final radius $r$ is $\infty$ because the satellite is far away from the earth and out of its gravitational field.

So, put $r=\infty$ to get the total work required to propel the rocket

  $W_{R\to \infty }=GmM\left(\frac{1}{R}-0\right)=\frac{GmM}{R}$

Now, equate the total work required to the amount of initial kinetic energy supplied to the rocket to write

  $\frac{1}{2}m{v}_e^2=\frac{GmM}{R}$

  $\Rightarrow v_e^2=\frac{2GM}{R}$

  $v_e=\sqrt{\frac{2GM}{R}}$

Therefore, the escape velocity ( $v_e$ ) on the planet is given by $\sqrt{\frac{2GM}{R}}$ , a quantity independent of the mass $m$ of the rocket (the object being propelled).