91影视

The given data is listed below:

• The distanceabove the surface of a planet is $\mathrm{d}=5.75脳{10}^5鈥�\mathrm{m}$ .
• The mass of the landing craft is $\mathrm{M}=12500鈥�\mathrm{kg}.$
• The period of the orbit is$\mathrm{t}=5800鈥�\mathrm{s}$.
• The diameter of the planet is$\mathrm{D}=9.60脳{10}^6鈥�\mathrm{m}.$.
• The mass of the astronaut is$\mathrm{m}=85.6鈥�\mathrm{kg}.$ .

In order to find the weight of the astronaut as he steps out of the planet's surface, the relation of gravitational force, as well as centripetal force will be applied.

The equation of speed of the craft can be written as,

$$\begin{gathered} \mathrm{v}=\frac{2\mathrm{蟺}(\mathrm{R}+\mathrm{d})}{\mathrm{t}} \\ \mathrm{v}=\frac{2\mathrm{蟺}\left({\displaystyle \frac{\mathrm{D}}{2}}+\mathrm{d}\right)}{\mathrm{t}} \end{gathered}$$

Substitute the values in the above equation.

$$\begin{gathered} \mathrm{v}=\left(\frac{2\mathrm{蟺}\left({\displaystyle \frac{9.60脳{10}^6鈥�\mathrm{m}}{2}}+5.75脳{10}^5鈥�\mathrm{m}\right)}{5800\mathrm{m}}\right) \\ \mathrm{v}=5822.77鈥�\mathrm{m}/\mathrm{s} \end{gathered}$$

The equation to find the mass of the planet can be written as,

$$\begin{gathered} {\mathrm{F}}_{\mathrm{G}}={\mathrm{F}}_{\mathrm{C}} \\ \frac{{\mathrm{Gm}}_{\mathrm{p}}\mathrm{m}}{{\left({\displaystyle \frac{\mathrm{D}}{2}}+\mathrm{d}\right)}^2}=\frac{{\mathrm{mv}}^2}{\left({\displaystyle \frac{\mathrm{D}}{2}}+\mathrm{d}\right)} \\ {\mathrm{m}}_{\mathrm{p}}=\frac{{\mathrm{v}}^2\left(\frac{\mathrm{D}}{2}+\mathrm{d}\right)}{\mathrm{G}} \end{gathered}$$

Here, Gis the universal gravitational constant whose value is$(6.67脳{10}^{-11})鈥�{\mathrm{m}}^3/\mathrm{kg}鈰�{\mathrm{s}}^2),{\mathrm{F}}_{\mathrm{G}}$ is the gravitational force, ${\mathrm{F}}_{\mathrm{C}}$and is the centripetal force.

Substitute the values in the above equation.

$$\begin{gathered} {\mathrm{m}}_{\mathrm{p}}=\left(\frac{\left(5822.77\mathrm{m}/{\mathrm{s}}^2\right)\left({\displaystyle \frac{9.60脳{10}^6\mathrm{m}}{2}}+5.75脳{10}^5\mathrm{m}\right)}{\left(6.67脳{10}^{-11}{\mathrm{m}}^3/\mathrm{kg}.{\mathrm{s}}^2\right)}\right) \\ {\mathrm{m}}_{\mathrm{p}}=2.73脳{10}^{24}\mathrm{kg} \end{gathered}$$

The equation of gravitational acceleration can be written as,

$$\begin{gathered} \mathrm{g}=\frac{{\mathrm{Gm}}_{\mathrm{P}}}{{\mathrm{R}}^2} \\ =\frac{{\mathrm{Gm}}_{\mathrm{P}}}{{\left({\displaystyle \frac{\mathrm{D}}{2}}\right)}^2} \end{gathered}$$

Here, Gis the universal gravitational constant,${\mathrm{m}}_{\mathrm{p}}$is the mass of the planet whose value is$(2.73脳{10}^{2}4鈥�\mathrm{kg}).$ .

Substitute the values in the above equation.

$$\begin{gathered} \mathrm{g}=\frac{(6.67脳{10}^{-11})鈥�{\mathrm{m}}^3/\mathrm{kg}鈰�{\mathrm{s}}^2\left)\right(2.73脳{10}^{24}\mathrm{kg})}{{\left({\displaystyle \frac{9.60脳{10}^6鈥�\mathrm{m}}{2}}\right)}^2} \\ \mathrm{g}=7.9鈥�\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

The equation of weight can be written as,

$\mathrm{W}=\mathrm{mg}$

Substitute the values in the above equation.

$$\begin{gathered} \mathrm{W}=(85.6鈥�\mathrm{kg})(7.9鈥�\mathrm{m}/{\mathrm{s}}^2) \\ \mathrm{W}=676.24鈥�\mathrm{N} \end{gathered}$$

Thus, the required weight of astronaut is $676.24\mathrm{N}$.