Question

Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earth’s mass is 5.98 x 10^24 kg and is concentrated at its center. Take the radius of the earth to be 6.37 x 10^6 m and G = 6.67 x 10^11 Nm2/kg2 .

Answer 1

Explanation:

In order to calculate the work from r = a to r = b we only need to convert 1000 km into 1,000,000 m.

Since, the given data is as follows.

           G = $6.67 \times 10^{-11} Nm/kg^{2}$

        $m_{1} = 5.98 \times 10^{24}$

        $m_{2}$ = 1000 kg

           a = $6.37 \times 10^{6} m$

           b = $6.37 \times 10^{6} + 1,000,000$

              = $7.37 \times 10^{6} m$

Now, we will calculate the work as follows.

              W = $Gm_{1}m_{2} [\frac{1}{a} - \frac{1}{b}]$

                   = $6.67 \times 10^{-11} \times 5.98 \times 10^{24} \times 1000 [\frac{1}{6.37 \times 10^{6}} - \frac{1}{7.37 \times 10^{6}}]$

                   = $8.50 \times 10^{9} J$

thus, we can conclude that work required to launch the satellite is $8.50 \times 10^{9} J$.